Isomorphism of C(x)-axa^-1 Function in Group G

[kathrynag]

Homework Statement

Let G be any group and let a be a fixed element of G. Define a function $$c_{a}$$:G-->G by $$c_{a}$$(x)=ax$$a^{-1}$$ for all x in G. Show that c is an isomorphism

The Attempt at a Solution

Need to show 1-1, onto and c(ab)=c(a)c(b)
I guess my biggest problem is starting because I get to c(a)=c(b) for 1-1 and don't know what c(a) is.

 

[micromass]

I'll get you started on 1-1. So take $$c(x)=axa^{-1}$$.
Assume that c(x)=c(y). Then by definition

$$axa^{-1}=aya^{-1}$$

Do you see how to continue?

 

[kathrynag]

Ok we want to show x=y
So axa^-1=aya^-1
axa^-1a=aya^-1a
ax=ay
a^-1ax=a^-1ay
x=y, so 1-1

For onto we need to c(x)=axa^-1=y
We need to be able to solve for x, I think
y=axa^-1
ya=axa^-1a
ya=ax
a^-1ya=a^-1ax
a^-1ya=x

 

[micromass]

Yes!

And now you just need to show that c is a homomorphism...

 

[kathrynag]

c(ab)=abx(ab)^-1
=abxa^-1b^-1
A bit confused...

 

[micromass]

No. Let's get you started:

$$c(x)c(y)=(axa^{-1})(aya^{-1})$$

 

[kathrynag]

ax(a^-1a)(ya^-1)
axya^-1=c(xy)

 

[micromass]

Yes!