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 x B, with B a quasi-injective abelian group of odd order and either K = Q(8) (the quaternion group of order 8) or K = 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 characterisation 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.

Onderzoeksraad of Vrije Universiteit Brussel

Onderzoeksraad of Vrije Universiteit Brussel

Departamento de Matematica of the Universidade Federal of Paraiba, Brazil

Departamento de Matematica of the Universidade Federal of Paraiba, Brazil

CNPq (Brazil)

CNPq-Brazil

FAPESP-Brazil [Proc. 2008/57930-1]

FAPESP (Brazil)

Fonds voor Wetenschappelijk Onderzoek (Belgium)

Fonds voor Wetenschappelijk Onderzoek (Belgium)