Groups and group theory
Intro
See intro to groups first!
Definition
A group is an algebraic structure $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} G = (G, \star)$ consisting of a set $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} G$ and a binary operation $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \star$ satisfying:
(composition) $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \star$ is associative
(identity) exists an $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} e \in G$ such that for any $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} f \in G$ we have $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} e\star f = f\star e = f$
(inverses) for any $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} f \in G$ exists a $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} g \in G$ such that $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} f\star g = g\star f = e$
Notation + termiology
Properties