Critical maximal subgroups and conjugacy of supplements in finite soluble groups
-
Barbara Baumeister
und
Gil Kaplan
Abstract
Let G be a finite group with an abelian normal subgroup N. When does N have a unique conjugacy class of complements in G? We consider this question with a focus on properties of maximal subgroups. As corollaries we obtain Theorems 1.6 and 1.7 which are closely related to a result by Parker and Rowley on supplements of a nilpotent normal subgroup [3, Theorem 1]. Furthermore, we consider families of maximal subgroups of G closed under conjugation whose intersection equals
Dedicated to the memory of Wolfgang Gaschütz (11.6.1920–7.11.2016)
Acknowledgements
The authors are indebted to Professor Hermann Heineken for fruitful discussions which helped to improve this paper. We also thank Chris Parker for his comments, and the second author wishes to thank Project C13 within SFB 701 at the University of Bielefeld for its hospitality.
References
[1] K. Doerk and T. Hawkes, Finite Soluble Groups, De Gruyter Exp. Math. 4, Walter de Gruyter, Berlin, 1992. 10.1515/9783110870138Suche in Google Scholar
[2] W. Gaschütz, Über die Φ-Untergruppe endlicher Gruppen, Math. Z. 58 (1953), 160–170. 10.1007/BF01174137Suche in Google Scholar
[3] C. Parker and P. Rowley, A note on conjugacy of supplements in finite soluble groups, Bull. Lond. Math. Soc. 42 (2010), no. 3, 417–419. 10.1112/blms/bdq002Suche in Google Scholar
[4] J. S. Rose, A Course on Group Theory, Cambridge University Press, Cambridge, 1978. Suche in Google Scholar
[5] M. Suzuki, Group Theory. I, Grundlehren Math. Wiss. 247, Springer, Berlin, 1982. 10.1007/978-3-642-61804-8Suche in Google Scholar
[6] M. Suzuki, Group Theory. II, Grundlehren Math. Wiss. 248, Springer, Berlin, 1986. 10.1007/978-3-642-86885-6Suche in Google Scholar
[7] D. L. Winter, The automorphism group of an extraspecial p-group, Rocky Mountain J. Math. 2 (1972), no. 2, 159–168. 10.1216/RMJ-1972-2-2-159Suche in Google Scholar
© 2017 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Irreducible representations of simple algebraic groups in which a unipotent element is represented by a matrix with a single non-trivial Jordan block
- Sandwich classification for GLn(R), O2n(R) and U2n(R,Λ) revisited
- Critical maximal subgroups and conjugacy of supplements in finite soluble groups
- Finite $\smash{p}$\hbox{-}groups of conjugate type {1,p3}
- Normal subgroups in limit groups of prime index
- Uncountable locally free groups and their group rings
- Full box spaces of free groups
- Almost congruence extension property for subgroups of free groups
- Solving equations of length seven over torsion-free groups
- On the structure of virtually nilpotent compact p-adic analytic groups
Artikel in diesem Heft
- Frontmatter
- Irreducible representations of simple algebraic groups in which a unipotent element is represented by a matrix with a single non-trivial Jordan block
- Sandwich classification for GLn(R), O2n(R) and U2n(R,Λ) revisited
- Critical maximal subgroups and conjugacy of supplements in finite soluble groups
- Finite $\smash{p}$\hbox{-}groups of conjugate type {1,p3}
- Normal subgroups in limit groups of prime index
- Uncountable locally free groups and their group rings
- Full box spaces of free groups
- Almost congruence extension property for subgroups of free groups
- Solving equations of length seven over torsion-free groups
- On the structure of virtually nilpotent compact p-adic analytic groups
