Group isomorphisms/Automorphism/Introduction/Section

Definition

Let ${}G$ and ${}H$ be groups. A bijective group homomorphism

$\varphi \colon G\longrightarrow H$

is called an isomorphism.

Bijective linear mappings are in particular group isomorphisms.

Definition

The groups ${}H$ and ${}G$ are called isomorphic, if there exists a group isomorphism

${}\varphi \colon G\rightarrow H$.

Lemma

Let ${}G$ and ${}H$ be groups, and let

$\varphi \colon G\longrightarrow H$

be a group isomorphism. Then also the inverse mapping

$\varphi ^{-1}\colon H\longrightarrow G,h\longmapsto \varphi ^{-1}(h),$

is a group isomorphism.

Proof

This follows from

${}{\begin{aligned}\varphi ^{-1}(h_{1}h_{2})&=\varphi ^{-1}{\left(\varphi (\varphi ^{-1}(h_{1}))\varphi (\varphi ^{-1}(h_{2}))\right)}\\&=\varphi ^{-1}{\left(\varphi {\left(\varphi ^{-1}(h_{1})\varphi ^{-1}(h_{2})\right)}\right)}\\&=\varphi ^{-1}(h_{1})\varphi ^{-1}(h_{2}).\end{aligned}}$

$\Box$

Example

We consider the additive group of the real numbers, that is ${}(\mathbb {R} ,0,+)$, and the multiplicative group of the positive real numbers, thus ${}(\mathbb {R} _{+},1,\cdot )$. Then the exponential function

$\exp \colon \mathbb {R} \longrightarrow \mathbb {R} _{+},x\longmapsto \exp(x),$

is a group isomorphism. This rests on basic analytic properties of the exponential function. The homomorphism property is just a reformulation of the functional equation

${}\exp(x+y)=e^{x+y}=e^{x}e^{y}=\exp(x)\exp(y)\,.$

The injectivity of the mapping follows from the strict monotonicity, the surjectivity follows from the Intermediate value theorem. The inverse mapping is the natural logarithm, which is also a group isomorphism.

Isomorphic groups are equal with respect to their group-theoretic properties. An isomorphism of a group to itself is called automorphism. The set of all automorphisms on ${}G$ form, with the composition of mappings, a group, which is denoted by ${}\operatorname {Aut} \,G$ and which is called the automorphism group of ${}G$. Important examples of automorphisms are the so-called inner automorphisms.

Definition

Let ${}G$ be a group, and ${}g\in G$ be fixed. The mapping defined by ${}g$,

$\kappa _{g}\colon G\longrightarrow G,x\longmapsto gxg^{-1},$

is called an inner automorphism.

The mapping ${}\kappa _{g}$ is also called the conjugation with ${}g$. If ${}G$ is a commutative Gruppe, then, because of ${}gxg^{-1}=xgg^{-1}=x$, the identity is the only inner automorphism. Therefore, this concept is only interesting for non-commutative groups.

Lemma

An inner automorphism is indeed an automorphism. The assignment

$G\longrightarrow \operatorname {Aut} \,G,g\longmapsto \kappa _{g},$

is a

group homomorphism.

Proof

We have

${}\kappa _{g}(xy)=gxyg^{-1}=gxg^{-1}gyg^{-1}=\kappa _{g}(x)\kappa _{g}(y)\,,$

so that ${}\kappa _{g}$ this is a group homomorphism. We have

${}\kappa _{g}(\kappa _{h}(x))=\kappa _{g}(hxh^{-1})=ghxh^{-1}g^{-1}=ghx(gh)^{-1}=\kappa _{gh}\,.$

This implies, on one hand, that

${}\kappa _{g^{-1}}\circ \kappa _{g}=\kappa _{g^{-1}g}=\operatorname {Id} _{G}\,;$

therefore, ${}\kappa _{g}$ is bijective and an automorphism. On the other hand, this implies that the total mapping ${}\kappa$ is a group homomorphism.

$\Box$

Example

For a fixed invertible matrix ${}B\in \operatorname {GL} _{n}\!{\left(K\right)}$, the conjugation

$\kappa _{B}\colon \operatorname {GL} _{n}\!{\left(K\right)}\longrightarrow \operatorname {GL} _{n}\!{\left(K\right)},M\longmapsto BMB^{-1},$

is just the mapping that assigns, to a describing matrix ${}M$ of a linear mapping with respect to a basis, the describing matrix with respect to a new basis.