Definition
A commutative group is a group whose operation is commutative.
Discussion
The following are equivalent sufficient conditions for commutativity:
$% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \forall x \in G : x^2 = 1$
$% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \forall x \in G : x = x^{-1}$
Proof:
(1 ⇒ 2)
$% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } a = a1 = aaa^{-1} = 1a^{-1} = a^{-1}$
(2 ⇒ 1)
$% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } a^2 = aa = aa^{-1} = 1$
(2 ⇒ commutative)
$% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } ab = a^{-1}b^{-1} = (ba)^{-1} = ba$