The relationship between inner automorphisms, commutativity, normality, and conjugacy.

An inner automorphism of a group $G$ is defined to be a function $f: G \to G$ such that for $x\in G$ $f(x) = a^{-1}xa.$

I have three somewhat broad questions about this:

1. Why is it related to the notion of an automorphism. That is, what about this says 'structure preserving'?
2. I have heard inner automorphisms say something about the degree to which commutativity is upheld in a group. How is this specifically?
3. How does this relate to the notions of normality conjugacy of a group?

Sorry if these questions are too broad.

• For a quick answer to #2, we have the following theorem: $G/Z(G)\cong Inn(G)$, where $Inn(G)$ is the group of inner automorphisms on $G$. Remember that if $a\in Z(G)$, then the $f$ in your question is the identity map. – luke May 3 '13 at 3:15
• $f(x)f(y)=a^{-1}xaa^{-1}ya=a^{-1}xya=f(xy)$. Also $f(1)=1$. Hence it is structure preserving. – vadim123 May 3 '13 at 3:17
• Thanks Zach, that makes sense, and thanks vadim, that one I should have seen! – user71443 May 3 '13 at 3:21

1 Answer

1. Inner automorphisms are exactly those automorphisms of the form you mention: $\theta_g:G\rightarrow G$ by $\theta_g(x)= g^{-1}xg$. It's easy to check that these are automorphisms: $g^{-1}xg=g^{-1}yg\Rightarrow x=y$ and $g^{-1}\left(gxg^{-1}\right)g=x$, so $\theta_g$ is bijective, and $\left(g^{-1}xg\right)\left(g^{-1}yg\right)=g^{-1}\left(xy\right)g$, so $\theta_g$ is a homomorphism. Thus $\theta_g$ is an isomorphism from the group to itself, which means that it's an automorphism. The set of automorphisms represents symmetries within the group. You can think of inner automorphisms as being intuitively a bit like a "change of basis" within the group.

2. A group has no non-trivial inner automorphisms if and only if it is abelian. This is easy to prove: try it!

3. Normal subgroups are exactly those subgroups which are fixed by every inner automorphism. This is similar to the concept of invariant subspaces in linear algebra. Note that normal subgroups only need be fixed set-wise under all inner automorphisms, rather than element-wise. In other words, given a $x$ in a normal subgroup $N$ and an inner automorphism $\theta_g$, we have only that $\theta_g(x)\in N$, not $\theta_g(x)=x$. Conjugacy classes of a group $G$ are equivalence classes of elements of $G$ under the conjugacy relation - that is, $x\sim y$ if and only if there exists a $g\in G$ such that $\theta_g(x)=y$. It is not difficult to see that any normal subgroup must be a disjoint union of conjugacy classes; this is the relationship between normality and conjugacy.