Can an outer automorphism fix the center of a group?

Question

I have just learned about inner and outer automorphisms, and I'm having a hard time visualizing the outer automorphism as quotient classes. Since the inner automorphisms fix the center, does it mean that any automorphism that fixes the center is necessarily an inner automorphism?

Answer 1

No, this is not the case: we can have an automorphism of a group which is not an inner automorphism but which nonetheless fixes the center pointwise - even when the center is nontrivial.

(As Mark Bennet observes, by contrast every automorphism whatsoever fixes the center in the weak sense that it restricts to a self-bijection of the center, but pointwise-fixing is more complicated. In particular, any nontrivial automorphism of an abelian group does not fix the center - which is after all the whole group - pointwise.)

For example, let $G$ be a group with trivial center which has some automorphism $\alpha$ which is not inner, and let $A$ be any nontrivial abelian group. Consider the group $G\times A$. The center of $G\times A$ is exactly the image of the map $A\rightarrow G\times A: a\mapsto (e_G, a)$ and so is nontrivial, and the automorphism defined by $$\hat{\alpha}: G\times A\rightarrow G\times A: (g,a)\mapsto (\alpha(g),a)$$ fixes that center. However, $\hat{\alpha}$ is not an inner automorphism of $G\times A$ since $\alpha$ wasn't an inner automorphism of $G$.

Answer 2

The centre is a characteristic subgroup, and is fixed by all automorphisms. That in itself is pretty much a tautology. But note the fact that the centre has a unique definition/property, even though there may be other subgroups isomorphic to the centre as abstract groups. And that is a property preserved by homomorphism.

Note therefore that the image of the centre of a group under any homomorphism must be commutative and contained within the centre of the image, and it is easy to show from this that an automorphism must fix the centre. Note further that this does not mean that an automorphism necessarily fixes the individual elements of the centre.