Group Isomorphisms

Isomorphism: $\forall a,b \in G, \varphi: G\to G', \varphi(ab) = \varphi(a)\varphi(b) \implies \varphi$ is bijective, then $\varphi$ is a isomorphism.

• $G$ is isomorphic to $G'$ note as $G\cong G'$

• Any group is isomorphic to itself (reflexive)

• $G\cong G' \implies G' \cong G$ (symmetric)

• $G\cong H, H\cong F \implies G \cong F$ (transitive)

• a bijective function $\varphi : G \to G$ is isomorphism $\implies \varphi$ is an automorphism of $G$

  • the set of all automorphisms of $G$ denote as $Aut(G)$

• any isomorphism is a homomorphism (but not vice versa)

• $G\cong \Z^n$, we define its rank $rk(G) = n$

Prove Structure for isomorphism:

1. Define $\varphi: G\to G'$, $\forall a,b\in G, \varphi(ab) = \varphi(a)\varphi(b)$
2. $\varphi$ is bijective

Example:

1. Prove $(\R, 0, +)\cong (\R^+, 1, \cdot)$
   1. Let $\varphi: \R \to \R^+$ where $\varphi(x) = e^x, \varphi(x+y) = e^{x+y}=e^xe^y=\varphi(x)\varphi(y)$
   2. let $x,y \in \R$ s.t. $e^x = e^y$ then $x = y \implies$ injective
   3. let $y \in \R^+,\exists \ln(y) \in \R, \varphi(\ln(y)) = y \implies$ surjective
   4. then $\varphi$ is isomorphism and $(\R, 0, +)\cong (\R^+, 1, \cdot)$

Isomorphism Property

let $\varphi: G\to \overline{G}$ be isomorphism :

• $\forall g\in G, ord(g) = ord(\varphi(g))$
• $\phi$ carries the identity of $G$ to the identity of $\overline{G}$
• $G$ is cyclic $\iff$ $\overline{G}$ is cyclic (i.e. $G = \langle g\rangle \iff \overline{G} = \langle \varphi(g)\rangle$)
• $G$ is abelian $\iff$ $\overline{G}$ is abelian (i.e For any elements $a$ and $b$ in $G$, $a$ and $b$ commute if and only if $\phi(a)$ and $\phi(b)$ commute)
• $\forall g\in G, C_G(g) \cong C_{\overline{G}}(\varphi(g))$
• If $G$ is finite, then $G$ and $\overline{G}$ have exactly the same number of elements of every order
• if $K \leq G$, then $\phi(K) \leq \overline{G}$

If such $\varphi_M(A) = MAM^{-1}$, we call $\varphi_M$ conjugation by $M$.

Cayley's Theorem: Every (finite) group is isomorphism to a group of permutation

Cayley's Theorem

Every (finite) group is isomorphism to a group of permutation

Automorphism

An isomorphism $\varphi: G\to G$ is called an automorphism of $G$.

Let $G$ be a group, $a \in G$. The function $\phi_a, s.t., \forall x\in G, \phi_a(x) = axa^{-1}$ is called the inner automorphism of $G$ induced by $a$.

The set of automorphisms of $G$ is denoted by $Aut(G)$ and the set of inner automorphisms of $G$ is denoted by $Inn(G)$. They are both groups under operation of composition.

For every positive integer $n$, $Aut(Z_n) \cong U(n)$