Question:Normal subgroup inner automorphism significance

Q: '''I have seen the definition of normal subgroup as invariance under conjugations, but then read something about these conjugations also being called inner automorphisms. Does this have any significance?'''

A: That the conjugation operations are inner automorphisms, and that invariance under these is equivalent to being normal, is very significant. This is because inner automorphisms are automorphisms, so they preserve the group structure. This is one explanation for why normality is strongly join-closed (the subgroup generated by a bunch of normal subgroups is normal), normality is centralizer-closed (the centralizer of a normal subgroup is normal), and normality is commutator-closed (the commutator of two normal subgroups is normal). It also explains why characteristic subgroups are normal, which explains why subgroup-defining functions such as the center, commutator subgroup, socle, and Frattini subgroup are all normal subgroups.