Geodesic
EXTENSION OF AUTOMORPHISMS OF SUBGROUPS
Glasgow mathematical journal, Tome 54 (2012) no. 2, pp. 371-380

Voir la notice de l'article provenant de la source Cambridge University Press

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.
DOI : 10.1017/S0017089512000031
Mots-clés : Primary 20D45, Secondary 20F28 
BASTOS, G. G.; JESPERS, E.; JURIAANS, S. O.; SILVA, A. DE A. E. EXTENSION OF AUTOMORPHISMS OF SUBGROUPS. Glasgow mathematical journal, Tome 54 (2012) no. 2, pp. 371-380. doi: 10.1017/S0017089512000031
@article{10_1017_S0017089512000031,
     author = {BASTOS, G. G. and JESPERS, E. and JURIAANS, S. O. and SILVA, A. DE A. E},
     title = {EXTENSION {OF} {AUTOMORPHISMS} {OF} {SUBGROUPS}},
     journal = {Glasgow mathematical journal},
     pages = {371--380},
     year = {2012},
     volume = {54},
     number = {2},
     doi = {10.1017/S0017089512000031},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089512000031/}
}
TY  - JOUR
AU  - BASTOS, G. G.
AU  - JESPERS, E.
AU  - JURIAANS, S. O.
AU  - SILVA, A. DE A. E
TI  - EXTENSION OF AUTOMORPHISMS OF SUBGROUPS
JO  - Glasgow mathematical journal
PY  - 2012
SP  - 371
EP  - 380
VL  - 54
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.1017/S0017089512000031/
DO  - 10.1017/S0017089512000031
ID  - 10_1017_S0017089512000031
ER  - 
%0 Journal Article
%A BASTOS, G. G.
%A JESPERS, E.
%A JURIAANS, S. O.
%A SILVA, A. DE A. E
%T EXTENSION OF AUTOMORPHISMS OF SUBGROUPS
%J Glasgow mathematical journal
%D 2012
%P 371-380
%V 54
%N 2
%U http://geodesic.mathdoc.fr/articles/10.1017/S0017089512000031/
%R 10.1017/S0017089512000031
%F 10_1017_S0017089512000031

[1] 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. Google Scholar | DOI

[2] 2.Bertholf, D. and Walls, G., Finite quasi-injective groups, Glasgow Math. J. 20 (1970), 29–33. Google Scholar | DOI

[3] 3.Eilenberg, S. and Moore, J. C., Foundations of relative homological algebra, Mem. Amer. Math. Soc. 55 (1965), 1–10. Google Scholar

[4] 4.Fuchs, L., Infinite abelian groups I (Academic Press, New York, 1970). Google Scholar

[5] 5.Gorenstein, D., Finite Groups (Harper & Row, New York – London, 1968). Google Scholar

[6] 6.Nogin, M., A short proof of Eilenberg and Moore's theorem, Cent. Eur. J. Math. 5 (1) (2007), 201–204. Google Scholar

[7] 7.Passman, D., Permutation groups (W. A. Benjamin Inc., New York, 1968). Google Scholar

[8] 8.Robinson, D. J. A., A Course in the theory of groups, second ed. (Springer-Verlag, New York, 1996). Google Scholar | DOI

[9] 9.Tomkinson, M. J., Infinite quasi-injective groups, Proc. Edinburgh Math. Soc. 31 (1988), 249–259. Google Scholar | DOI

Cité par Sources :