2017
DOI: 10.1134/s0037446617030107
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On the pronormality of subgroups of odd index in finite simple symplectic groups

Abstract: A subgroup H of a group G is said to be pronormal in G if H and H g are conjugate in H, H g for every g ∈ G. In [Sib. Math. J. 2012. Vol. 53, no. 3], the following conjecture was formulated by Evgeny P. Vdovin and the third author.Conjecture. All subgroups of odd indices are pronormal in all finite simple groups.The conjecture was verified by authors for many families of finite simple groups in [Sib. Math. J. 2015. Vol. 56, no. 6]. Namely, the following theorem was proved.Theorem. All subgroups of odd indices… Show more

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Cited by 14 publications
(17 citation statements)
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“…Now it is easy to see that therefore, H 1 × M 2 is a non-pronormal subgroup of odd index in M 1 × M 2 . So, H 1 × M 2 is a nonpronormal subgroup of odd index in G. Thus, we proved the following theorem (see [13,Theorem 1]). Theorem 6.…”
Section: Simple Symplectic Groups Containing Non-pronormal Subgroups mentioning
confidence: 77%
See 1 more Smart Citation
“…Now it is easy to see that therefore, H 1 × M 2 is a non-pronormal subgroup of odd index in M 1 × M 2 . So, H 1 × M 2 is a nonpronormal subgroup of odd index in G. Thus, we proved the following theorem (see [13,Theorem 1]). Theorem 6.…”
Section: Simple Symplectic Groups Containing Non-pronormal Subgroups mentioning
confidence: 77%
“…Proposition 8. (see [13,Lemma 15] and [5, Lemma 9]) Let Q be a subgroup of odd index in a group L = L 1 × L 2 × . .…”
Section: Classification Of Simple Symplectic Groups In Which the Subgmentioning
confidence: 99%
“…Moreover, in [13,14], we proved that if q Á˙3 .mod 8/ and n 2 ¹2 m ; 2 m .2 2k C 1/ j m; k 2 N [ ¹0ºº;…”
Section: /mentioning
confidence: 82%
“…In 2012, E. Vdovin and the third author [20] proved that Hall subgroups (when they exist) are pronormal in all simple groups and, guided by the analysis in their proof, they conjectured that any subgroup of odd index of a simple group is pronormal in this group. This conjecture was disproved in [12,13]. Precisely, if q Á˙3 .mod 8/ and n 6 2 ¹2 m ; 2 m .2 2k C 1/ j m; k 2 N [ ¹0ºº, then the simple symplectic group PSp 2n .q/ contains a non-pronormal subgroup of odd index.…”
Section: Introductionmentioning
confidence: 99%
“…In 2012, E. Vdovin and the second author [15] proved that the Hall subgroups are pronormal in all simple groups and, guided by the analysis in their proof, they conjectured that any subgroup of odd index of a simple group is pronormal in this group. This conjecture was disproved in [9,10]. In [8,9,10,11], finite simple groups in which all the subgroups of odd index are pronormal were studied.…”
Section: Introductionmentioning
confidence: 99%