# $\operatorname{Aut}_c(G)$ vs "$\operatorname{Inn}_c(G)$"

As an application of the correspondence theorem, time ago I saw that the group of the inner automorphisms which commute with every automorphism -say $$\operatorname{Inn}_c(G)$$- is isomorphic to $$H(G)/Z(G)$$, where: $$H(G)=\{g \in G \mid \varphi(g)g^{-1} \in Z(G), \forall \varphi \in \operatorname{Aut}(G) \}\supseteq Z(G)$$ More recently, I've come across with the group of the automorphisms which commute with every inner automorphism. This central automorphism group, $$\operatorname{Aut}_c(G)$$, has much more literature than $$\operatorname{Inn}_c(G)$$, which hasn't got any indeed (at least to my knowledge). Is $$\operatorname{Inn}_c(G)$$ of any interest and hence mentioned somewhere?

Good question. The group of automorphisms you are looking at is basically $$\rm{Inn(G)} \cap Z(\rm{Aut}(G))$$, where the center is taken in $$\rm{Aut(G)}$$. These are called autocentral automorphisms. These have been studied in more detail amongst others by (the students of) prof. M.R. Moghaddam (see for example here).

• Thank you for the link. So, seemingly the following chain holds: $L(G)\le Z(G)\le H(G)\le G$, thus hinting for possibly further quotients than $\operatorname{Inn}(G)\cong G/Z(G)$ and $\operatorname{Inn}_c(G)\cong H(G)/Z(G)$? (+1) and accepted.
– user1007416
Mar 8, 2022 at 12:52
• Yes that is correct. And both centralizers $\rm{Aut}_c(G)=C_{\rm{Aut}(G)}(\rm{Inn}(G))$ and $\rm{Inn}_c(G)=C_{\rm{Inn}(G)}(\rm{Aut}(G))$ are interesting objects to study. Mar 8, 2022 at 14:25