The reduction theorem for relatively maximal subgroups

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Let be a class of finite groups closed under taking subgroups, homomorphic images and extensions. It is known that if A is a normal subgroup of a finite group G then the image of an-maximal subgroup H of G in G/A is not, in general,-maximal in G/A. We say that the reduction-Theorem holds for a finite group A if, for every finite group G that is an extension of A (i.e. contains A as a normal subgroup), the number of conjugacy classes of-maximal subgroups in G and G/A is the same. The reduction-Theorem for A implies that HA/A is-maximal in G/A for every extension G of A and every-maximal subgroup H of G. In this paper, we prove that the reduction-Theorem holds for A if and only if all-maximal subgroups of A are conjugate in A and classify the finite groups with this property in terms of composition factors.

Original languageEnglish
Article number2150001
JournalBulletin of Mathematical Sciences
Issue number1
Publication statusPublished - 1 Apr 2022


  • -maximal subgroup
  • -submaximal subgroup
  • Complete class
  • finite simple group




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