EXTENSION OF AUTOMORPHISMS OF SUBGROUPS

Author(s)

Bastos, G. G. ^[1] ; Jespers, E. ^[2] ; Juriaans, S. O. ^[3] ; Silva, A. De A. E. ^[4]

Affiliation

^[1] Univ Fed Ceara, Dept Matemat, BR-60455760 Fortaleza, Ceara - Brazil

^[2] Vrije Univ Brussel, Dept Math, B-1050 Brussels - Belgium

^[3] Univ Sao Paulo, Inst Matematica & Estat, BR-05311970 Sao Paulo - Brazil

^[4] Univ Fed Paraiba, Dept Matemat, BR-58051900 Joao Pessoa, Paraiba - Brazil

• Document type: Journal article
• Source: Glasgow Mathematical Journal; v. 54, n. 2, p. 371-380, MAY 2012.
• Web of Science Citations: 0

Abstract

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. (AU)