Inner Automorphism: Clarifying Notation Confusion

[Gale]

I'm working some homework where we're introduced to inner automorphisms. I have that G is a group. Given a fixed g in G, and f_g:G → G, f_g(x)= gxg^-1.

I'm just a little confused about the notation. Are both g and x in G? I've been reading through the wiki http://en.wikipedia.org/wiki/Inner_automorphism and that seems to be the case, however when I'm working on proofs I wasn't sure which variable I was supposed to be working with.

For instance, if I'm proving surjectivity, (or injectivity or bijectivity) I would want to show f(x₁)=f(x₂), however I would assume g stays fixed, correct? But when I'm proving f has an inverse, I would prove that (f_g)^-1=f_g^-1, which is using a different fixed g in G? Similarly, The wiki says that when G is abelian, the inner automorphism contains only the identity, but if I wanted to prove that, would I work with two different g's or two different x's?

I have a vague idea of how this relates to permutations, but it's not sinking in and my hang ups about the notation is preventing me from understanding a lot of what I'm reading. f is a permutation function? Or is G a set of permutations? Actually, now I'm confusing myself even worse. Could someone clarify for me?

 

[micromass]

I'm working some homework where we're introduced to inner automorphisms. I have that G is a group. Given a fixed g in G, and f_g:G → G, f_g(x)= gxg^-1.

I'm just a little confused about the notation. Are both g and x in G?

Yes.

I've been reading through the wiki http://en.wikipedia.org/wiki/Inner_automorphism and that seems to be the case, however when I'm working on proofs I wasn't sure which variable I was supposed to be working with.

For instance, if I'm proving surjectivity, (or injectivity or bijectivity) I would want to show f(x₁)=f(x₂), however I would assume g stays fixed, correct?

Correct. Every g induces an inner automorphism $f_g$. The function $f_g$ is a function of x, so only x is variable.

But when I'm proving f has an inverse, I would prove that (f_g)^-1=f_g^-1, which is using a different fixed g in G?

Yes.

Similarly, The wiki says that when G is abelian, the inner automorphism contains only the identity, but if I wanted to prove that, would I work with two different g's or two different x's?

You need to prove that each function $f_g$ is equal to the identity.

I have a vague idea of how this relates to permutations, but it's not sinking in and my hang ups about the notation is preventing me from understanding a lot of what I'm reading. f is a permutation function? Or is G a set of permutations? Actually, now I'm confusing myself even worse. Could someone clarify for me?

A permutation of a set X is by definition a bijection $g:X\rightarrow X$. Any inner automorphism $f_g$ is a permutation on G since it is bijective.