# Topic Category: Isomorphism of Groups

## Inner automorphism

Let G be a group and let . The function defined by is called the inner automorphism of G induced by a. We can easily prove that is actually a automorphism of G. Step 1 : If x = y then So, is well defined Step 2 : So, is one-one. Step 3 : For …

## Examples of Automorphism

Example 1 : Consider the mapping defined on by Clearly, is bijective Now, let & So, preserves the operation, so, is an automorphism in Geometrically, we can see this automorphism as composition of two automorphism, reflection through x-axis followed by reflection through the line y = x. Example 2 : Let G be a group …

## Automorphism

An isomorphism from a group G onto itself is called automorphism of G. For example : The function given is an automorphism of complex number under addition. The automorphism of are (a) (reflection in X-axis) (b) (reflection in Y-axis) (c) (reflection in the line y = x) (d) Where c is a non-zero real number. …

## Properties of Isomorphism

Theorem 1 : Properties of Isomorphisms Acting on Elements Suppose that is an isomorphism from a group G onto a group . Then 1. carries the identity of G to the identity of . 2. For every integer n and for every group element a in 3. For any elements a and b in G, …

## Cayley Theorem

Every group is isomorphic to group of permutations. Proof : To prove this, let G be any group. We must find a group of permutation that we believe is isomorphic to G. Since G is all we have to work with, we will have to use it to construct . For any g in G, …

## Examples of Isomorphism

Example 1 : Consider the groups is group of real numbers with addition operation and is group of positive real number with multiplication operation Are these group isomorphic to each other? Let us define a map f from to as Certainly, f is a well defined function from to Suppose then x = y So, …

## Introduction to Isomorphism

Before we formally define isomorphism, let us consider some subgroups. Consider a subgroup is a subgroup of set of non zero complex number equipped with multiplication operation. is a cyclic subgroup whose generator is i So, Now, let us consider a cyclic subgroup of generated by So, Now, let us draw multiplication table of both …