Extension of Automorphisms of Subgroups

Let $G$ be a group such that, for any subgroup $H$ of $G$, every automorphism of $H$ can be extended to an automorphism of $G$. Such a group $G$ is said to be of injective type. The finite abelian groups of injective type are precisely the quasi injective groups. We prove that a finite non-abelian group $G$ of injective type has even order. If, furthermore, $G$ is also quasi-injective,
then we prove that $G=K \times B$, with $B$ a quasi-injective abelian group of odd order and
either $K=Q_{8}$ (the quaternion group of order $8$) or $K=\mbox{Dih}(A)$, a dihedral group on a
quasi-injective abelian group $A$ of odd order coprime with the order of $B$. We give a description of the supersoluble finite groups of injective type whose Sylow $2$-subgroup are abelian showing that
these groups are, in general, not quasi-injective. In particular, the characterization of such groups is reduced to that of finite $2$-groups that are of injective type. We give several restrictions on the latter. We also show that the alternating group $A_5$ is of injective type but that the binary icosahedral group $SL(2,5)$ is not.

Jaar van publicatie:2012

Trefwoorden:automorphism, group, lifting