On central automorphisms of crossed modules

Abstract

A crossed module $(T,G,\partial)$ consist of a group homomorphism $\partial:T\rightarrow G$ together with an action $(g,t)\rightarrow{}^{\,g}t$ of $G$ on $T$ satisfying $\partial(^{\,g}t)=g\partial(t)g^{-1}$ and $\,^{\partial(s)}t=sts^{-1}$, for all $g\in G$ and $s,t\in T$. The term crossed module was introduced by J. H. C. Whitehead in his work on combinatorial homotopy theory. Crossed modules and its applications play very important roles in category theory, homotopy theory, homology and cohomology of groups, algebra, K-theory etc. In this paper, we define Adeny-Yen crossed module map and central automorphisms of crossed modules. If $C^*$ is the set of all central automorphisms of crossed module $(T,G,\partial)$ fixing $Z(T,G,\partial)$ element-wise, then we give a necessary and sufficient condition such that $C^*=I_{nn}(T,G,\partial).$ In this case, we prove $Aut_C(T,G,\partial)\cong Hom((T,G,\partial), Z(T,G,\partial))$. Moreover, when $Aut_C(T,G,\partial)\cong Z(I_{nn}(T,G,\partial)))$, we obtain some results in this respect.