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EXTENSION OF AUTOMORPHISMS OF SUBGROUPS

Published online by Cambridge University Press:  29 March 2012

G. G. BASTOS ,
E. JESPERS ,
S. O. JURIAANS  and
A. DE A. E SILVA
G. G. BASTOS
Affiliation:
Departamento de Matemática, Universidade Federal do Ceará, Campus do Pici, CEP 60455-760, Fortaleza, Ceará, Brazil e-mail: ggbastos@ufc.br
E. JESPERS
Affiliation:
Department of Mathematics, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussel, Belgium e-mail: efjesper@vub.ac.be
S. O. JURIAANS
Affiliation:
Instituto de Matemática e Estatística, Universidade de São Paulo, CP 66281, CEP 05311-970, São Paulo, Brazil e-mail: ostanley@ime.usp.br
A. DE A. E SILVA
Affiliation:
Departamento de Matemática, Universidade Federal da Paraíba, CEP 58051-900, João Pessoa, Pb, Brazil e-mail: andrade@mat.ufpb.br
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Abstract

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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 × B, with B a quasi-injective abelian group of odd order and either K = Q8 (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 A5 is of injective type but that the binary icosahedral group SL(2, 5) is not.

Keywords

Primary 20D45Secondary 20F28
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 54 , Issue 2 , May 2012 , pp. 371 - 380
Copyright
Copyright © Glasgow Mathematical Journal Trust 2012

References

REFERENCES

1.Azevedo, A. C. P, Bastos, G. G. and Juriaans, S. O., Extension of automorphisms of subgroups of abelian and polycyclic-by-finite groups, J. Algebr Appl. 6 (2) (2007), 315–322.CrossRefGoogle Scholar
2.Bertholf, D. and Walls, G., Finite quasi-injective groups, Glasgow Math. J. 20 (1970), 29–33.CrossRefGoogle Scholar
3.Eilenberg, S. and Moore, J. C., Foundations of relative homological algebra, Mem. Amer. Math. Soc. 55 (1965), 1–10.Google Scholar
4.Fuchs, L., Infinite abelian groups I (Academic Press, New York, 1970).Google Scholar
5.Gorenstein, D., Finite Groups (Harper & Row, New York – London, 1968).Google Scholar
6.Nogin, M., A short proof of Eilenberg and Moore's theorem, Cent. Eur. J. Math. 5 (1) (2007), 201–204.Google Scholar
7.Passman, D., Permutation groups (W. A. Benjamin Inc., New York, 1968).Google Scholar
8.Robinson, D. J. A., A Course in the theory of groups, second ed. (Springer-Verlag, New York, 1996).CrossRefGoogle Scholar
9.Tomkinson, M. J., Infinite quasi-injective groups, Proc. Edinburgh Math. Soc. 31 (1988), 249–259.CrossRefGoogle Scholar