Question

If C is a conjugacy class in G and f is an automorphism of G, prove that f (C) is also a conjugacy class of G.

Short answer

It is proved that $f\left(\mathrm{C}\right)$ is also a conjugacy class of G .

Step-by-step solution

Automorphism

An isomorphism mapping of a group G onto G it self is called an automorphism of G .

A mapping $f:G\to G$ is called an automorphism of G if

1. f is bijection

2. $f\left(ab\right)=f\left(a\right)f\left(b\right)$

Let C is a conjugacy class in G

Let f is an automorphism of G .

Therefore, f is one-one and onto.

Conjugacy class

Let G be a group and $a,b\in G$. We say that ais conjugate tobif there exist $x\in G$such that $\mathit{b}\mathbf{=}{\mathit{x}}^{\mathbf{-}\mathbf{1}}\mathit{a}\mathit{x}$.

The conjugacy class of an elementaconsists of all the elements in G that are conjugate toa.

Let, $b\in C\Rightarrow \left\{g^{-1}ag;g\in G\right\}$[by definition of conjugacy class]$\Rightarrow f\left(b\right)=f\left(g^{-1}ag\right)$

$\Rightarrow$ $=f\left(g^{-1}a\right)f\left(g\right)$ [ is automorphism ]

$\Rightarrow$ $=f\left(g^{-1}\right)f\left(a\right)f\left(g\right)$ [ is automorphism ]

$\Rightarrow$ $=f{\left(g\right)}^{-1}f\left(a\right)f\left(g\right)$ [ is onto]

Therefore,$f\left(a\right)$ is conjugate of $f\left(b\right)$ .

Here also f is automorphism

Therefore,$f\left(c\right)$is also a conjugacy class of G .