# About $E(G)$ for a finite $p$-group $G$

For any group $$G$$, the absolute center $$L(G)$$ of $$G$$ is defined as $$L(G) = \lbrace g\in G\mid \alpha(g)=g,\forall\alpha\in Aut(G) \rbrace$$, where $$Aut(G)$$ denote the group of all automorphisms of $$G$$. An automorphism $$\alpha$$ of $$G$$ is called an absolute central automorphism if $$g^{-1}\alpha(g)\in L(G)$$ for all $$g\in G$$. Let $$Var(G)$$ denote the group of all absolute central automorphisms of $$G$$. Let $$C_{Aut(G)}(Var(G))= \lbrace \alpha\in Aut(G)\mid\alpha\beta = \beta\alpha, \forall\beta \in Var(G)\rbrace$$ denote the centralizer of $$Var(G)$$ in $$Aut(G)$$. Let $$E(G)=[G,C_{Aut(G)}(Var(G))]=\langle g^{-1}\alpha(g)\mid g\in G, \alpha\in C_{Aut(G)}(Var(G))\rangle.$$ One can easily see that $$E(G)$$ is a characteristic subgroup of $$G$$ containing the derived group $$G^{\prime}=[G,Inn(G)]$$. Let $$G$$ be a finite $$p$$-group, how to calculate $$E(G)$$ in GAP?

• $E(G)$ is contained in the intersection of all kernels of homomorphisms $G\to L(G)$. Do they always coincide? – Ilya Bogdanov Apr 16 at 9:06