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