Question:Normal subgroup characteristic subgroup invariance confusion

See also Question:Normal subgroup characteristic subgroup relation.

Q: '''I was told by some people that normal subgroups are precisely the subgroups that are invariant, but the correct term for that seems to be characteristic subgroup. How do you explain this?'''

A: It depends on what sort of invariance is being sought. If we are looking for invariance under automorphisms of the group, then the correct notion is that of characteristic subgroup.

However, normality is more important in the following sense: when a group acts on a structure, we are interested in those subgroups of the group that are invariant under change of coordinates on the set, which means invariant under automorphisms of the set. Since the group itself acts by automorphisms, this in particular implies invariance under the action of the group. But when a group acts on a set, the induced action on itself is the group action by conjugation. Thus, all subgroups invariant in this manner must be normal; however, they need not be characteristic. Since most of the applications of groups to other areas of mathematics is via group actions, normality is the more important notion.