- #1

PingPong

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## Homework Statement

Let G be a group. We showed in class that the permutations of G which send products to products form a subgroup Aut(G) inside all the permutations. Furthermore, the mappings of the form [itex]\sigma_b(g)=bgb^{-1}[/itex] form a subgroup inside Aut(G) called the inner automorphisms and denoted Inn(g).

Prove that the inner automorphisms form a normal subgroup of Aut(G).

## Homework Equations

None

## The Attempt at a Solution

I attempted this problem one way, and my professor said I was going about it the wrong way - I did multiplications of the permutations and apparently I'm supposed to use function composition of the permutations. So this is what I have: I must show that, given a permutation [itex]\tau[/itex] that [itex]\tau \sigma_b \tau^{-1}[/itex] is in the set of inner automorphisms (at least, that's how we've been showing normality in class and my professor told me that this is at least correct).

So, I have

[tex]\tau\left(\sigma_b\left(\tau^{-1}\left(g\right)\right)\right)&=&\tau\left(b\tau^{-1}(g)b^{-1}\right)\\

&=&\tau(b)\tau\left(\tau^{-1}(g)\right)\tau(b^{-1})\\

&=&\tau(b)g\tau(b)^{-1}\\

&=&\sigma_{\tau(b)}(g)[/tex]

My question is, is [itex]\tau[/itex] equal to [itex]\tau^{-1}(g)[/itex] or is it [itex]\left(\tau(g)\right)^{-1}[/itex], or are these the same thing? I've done the problem assuming that [itex]\tau^{-1}[/itex] means [itex]\tau^{-1}(g)[/itex] and it seems to work, but I'm a bit uneasy about this. Can anybody confirm whether I've done this properly or not?