I have often found it useful to sit and contemplate what kinds of elements, subsets, or structures do the automorphisms of an object fix or permute. Sometimes the observations do not have immediate application, but they have often come back to help me later on. To help my future self, I'm asking about groups in particular.

What kind of elements, subsets, or structures do automorphisms of a group fix, permute, or preserve?

The examples I've compiled and some of their uses are listed below:

  • Subgroups (and subgroups of a fixed cardinality)
  • Normal subgroups
  • Maximal subgroups (implies that the intersection of all maximal subgroups is characteristic)
  • Minimal subgroups (implies that the subgroup they generate is a characteristic subgroup)
  • Sylow $p$-subgroups for any prime $p$
  • Subgroups of index $n$ for any $n\in\Bbb N$
  • Elements of order $n$ for any $n\in\Bbb N$
  • Elements in the center (implies the center is characteristic)
  • Commutators (implies the derived subgroup is characteristic)
  • Generating sets (this helps classify the automorphisms of a cyclic group)
  • Minimal generating sets
  • Nongenerators
  • Conjugacy classes (and conjugacy classes of a fixed cardinality) beautiful application
  • Centralizers
  • Normalizers

This is a fairly big list but is certainly far from exhaustive. And I feel like if I was exposed to new examples, I would see new connections that I haven't before.

Any addition to the list is much appreciated. Bonus points for an application of why the observation is useful (if I could give bonus points) .

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    $\begingroup$ To say an automorphism $\varphi$ of a group $G$ fixes a subgroup $H \subset G$ means that $\varphi(H) \subset H$. I think you mean preserve: the property of $H$ being a subgroup is preserved by the automorphism. $\endgroup$ – RghtHndSd Aug 31 '14 at 3:41
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    $\begingroup$ Sure. As for the question itself, an isomorphism of groups can be thought of as just relabeling the elements of the group. So anything defined solely in terms of group theory will be preserved. Other structures such as topology won't necessarily be preserved. $\endgroup$ – RghtHndSd Aug 31 '14 at 3:47
  • $\begingroup$ @RghtHndSd I'm aware. Do you have more examples? $\endgroup$ – Robert Wolfe Aug 31 '14 at 4:01
  • $\begingroup$ Nilpotent, solvable, finitely presented, finitely generated, supersolvable, polycyclic, virtually polycyclic, strongly polycyclic... Of course, listing out all the pages I can find on Wikipedia about group theory hardly seems like a good use of time. $\endgroup$ – RghtHndSd Aug 31 '14 at 15:04
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    $\begingroup$ I am surprised no one mentioned the most important (arguably even definitional) example of what group automorphisms preserve: relations between elements. This is true for any concrete algebraic structure. $\endgroup$ – whacka Sep 3 '14 at 21:50