Multiplicative notation (the usual): Additive notation:

Proof. Assume two identities $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } e, e'$. Then $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } e = ee' = e'$

Proof. Fix $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } f$, and assume two inverses $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } g, g'$. Then $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } g = g(fg') = (gf)g' = g'$.

$$\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } (ab)^n = \underbrace {ab \cdot ab \cdots ab}_{n \text{ times}} = (\underbrace {a \cdot a \cdots a}_{n \text{ times}}) \cdot (\underbrace {b \cdot b \cdots b}_{n \text{ times}}) = a^n b^n$$

Follows from axiom (3). Let $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } f=a$, then exists $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } a^{-1}$ such that $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } aa^{-1} = 1$. Since inverses are two-sided this implies both that $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } a^{-1}$ is an inverse of $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } a$ and that $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } a$ is an inverse of $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } a^{-1}$.

This can be shown directly; for fun, though, we will prove it using Cayley's isomorphism $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \phi$. First note that for $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } f, g$ bijective, we have $$\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } (f \circ g)^{-1} = g^{-1} \circ f^{-1}$$ So given $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } a,b \in G$ this gives $$\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \begin{align} (\phi(a) \circ \phi(b))^{-1} &= \phi(a)^{-1} \circ \phi(b)^{-1} \\ \phi(ab)^{-1} &= \phi(a)^{-1} \circ \phi(b)^{-1} \\ \phi((ab)^{-1}) &= \phi(a^{-1}) \circ \phi(b^{-1}) \\ \phi((ab)^{-1}) &= \phi(a^{-1}b^{-1}) \\ (ab)^{-1} &= a^{-1}b^{-1} \end{align}$$ Note that step $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } (2 \to 3)$ follows from the property of group isomorphisms that $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \phi(a)^{-1} = \phi(a^{-1})$.