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On the converse of Gaschütz’ complement theorem

  • Benjamin Sambale ORCID logo EMAIL logo
Published/Copyright: May 11, 2023
Journal of Group Theory
From the journal Journal of Group Theory Volume 26 Issue 5

Abstract

Let 𝑁 be a normal subgroup of a finite group 𝐺. Let N ≤ H ≤ G such that 𝑁 has a complement in 𝐻 and ( | N | , | G : H | ) = 1 . If 𝑁 is abelian, a theorem of Gaschütz asserts that 𝑁 has a complement in 𝐺 as well. Brandis has asked whether the commutativity of 𝑁 can be replaced by some weaker property. We prove that 𝑁 has a complement in 𝐺 whenever all Sylow subgroups of 𝑁 are abelian. On the other hand, we construct counterexamples if Z ⁢ ( N ) ∩ N ′ ≠ 1 . For metabelian groups 𝑁, the condition Z ⁢ ( N ) ∩ N ′ = 1 implies the existence of complements. Finally, if 𝑁 is perfect and centerless, then Gaschütz’ theorem holds for 𝑁 if and only if Inn ⁢ ( N ) has a complement in Aut ⁢ ( N ) .

Funding source: Deutsche Forschungsgemeinschaft

Award Identifier / Grant number: SA 2864/1-2

Award Identifier / Grant number: SA 2864/4-1

Funding statement: The work is supported by the German Research Foundation (SA 2864/1-2 and SA 2864/4-1).

Acknowledgements

Proposition 7 ii was found by Scheima Obeidi within the framework of her Master’s thesis written under the direction of the author. I appreciate some valuable comments of an anonymous referee.

  1. Communicated by: Andrea Lucchini

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Received: 2022-10-24
Revised: 2023-02-28
Published Online: 2023-05-11
Published in Print: 2023-09-01

© 2023 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Hochschild cohomology of symmetric groups and generating functions
  3. Stiefel–Whitney classes of representations of SL(2, 𝑞)
  4. A classification of finite primitive IBIS groups with alternating socle
  5. On the converse of Gaschütz’ complement theorem
  6. Conjugacy classes of maximal cyclic subgroups
  7. Morse boundaries of graphs of groups with finite edge groups
  8. The number of locally invariant orderings of a group
  9. On limits of betweenness relations
https://doi.org/10.1515/jgth-2022-0178

Articles in the same Issue

  1. Frontmatter
  2. Hochschild cohomology of symmetric groups and generating functions
  3. Stiefel–Whitney classes of representations of SL(2, 𝑞)
  4. A classification of finite primitive IBIS groups with alternating socle
  5. On the converse of Gaschütz’ complement theorem
  6. Conjugacy classes of maximal cyclic subgroups
  7. Morse boundaries of graphs of groups with finite edge groups
  8. The number of locally invariant orderings of a group
  9. On limits of betweenness relations
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