We prove that if A ≠ 1 is a subgroup of a finite group G and the order of an element in the centralizer of A in G is strictly larger (larger or equal) than the index [G:A], then A contains a non-trivial characteristic (normal) subgroup of G. Consequently, if A is a stabilizer in a transitive permutation group of degree m > 1, then exp (Z(A)) < m. These theorems generalize some recent results of Isaacs and the authors. © 2003 Hebrew University.
On subgroups containing non-trivial normal subgroups
LUCCHINI, ANDREA
2003
Abstract
We prove that if A ≠ 1 is a subgroup of a finite group G and the order of an element in the centralizer of A in G is strictly larger (larger or equal) than the index [G:A], then A contains a non-trivial characteristic (normal) subgroup of G. Consequently, if A is a stabilizer in a transitive permutation group of degree m > 1, then exp (Z(A)) < m. These theorems generalize some recent results of Isaacs and the authors. © 2003 Hebrew University.
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