International Journal of Group Theory
Volume:11 Issue: 1, Mar 2022

A subgroup A of a group G is called {\it seminormal} in G‎, ‎if there exists a subgroup B such that G=AB and AX~is a subgroup of G for every‎ ‎subgroup X of B‎. ‎The group G=G1G2⋯Gn with pairwise permutable subgroups G1‎,‎…‎,‎Gn such that Gi and Gj are seminormal in~GiGj for any i‎,‎j∈{1,…‎,‎n}‎, ‎i≠j‎, ‎is studied‎. ‎In particular‎, ‎we prove that if Gi∈F for all i‎, ‎then GF≤(G′)N‎, ‎where F is a saturated formation and U⊆F‎. ‎Here N and U‎~ ‎are the formations of all nilpotent and supersoluble groups respectively‎, ‎the F-residual GF of G is the intersection of all those normal‎ ‎subgroups N of G for which G/N∈F‎.

Let G be a group and ω(G)={o(g)|g∈G} be the set of element orders of G. Let k∈ω(G) and sk=|{g∈G|o(g)=k}|. Let nse(G)={sk|k∈ω(G)}. In this paper, we prove that if G is a group and G2(5) is the Chevalley simple group of type G2 over GF(5) such that nse(G)=nse(G2(5)), then G≅G2(5).

The spectrum of a finite group is the set of‎ ‎its element orders‎. ‎H is said to be a finite cover of G if G‎ ‎is a homomorphic image of H and H is finite‎. ‎The main aim of‎ ‎this article is to characterize the finite simple groups with no‎ ‎elements of order 10 by its spectrum among covers‎. ‎At the same‎ ‎time‎, ‎above simple groups are completely classified‎. ‎At last‎, ‎some‎ ‎results on the recognition by spectrum of above groups are also‎ ‎achieved‎.

A binary triply-even [98280,25,47104]2 code invariant under the sporadic simple group Co1 is constructed by adjoining the all-ones vector to the faithful and absolutely irreducible 24-dimensional code of length 98280. Using the action of Co1 on the code we give a description of the nature of the codewords of any non-zero weight relating these to vectors of types 2, 3 and 4, respectively of the Leech lattice. We show that the stabilizer of any non-zero weight codeword in the code is a maximal subgroup of Co1. Moreover, we give a partial description of the nature of the codewords of minimum weight of the dual code.

We study soluble groups G in which each subnormal subgroup H with infinite rank is‎ ‎commensurable with a normal subgroup‎, ‎i.e‎. ‎there‎ ‎exists a normal subgroup N such that H∩N has finite index‎ ‎in both H and N‎. ‎We show that if such a G is periodic‎, ‎then‎ ‎all subnormal subgroups are commensurable with a normal subgroup‎, ‎provided either the Hirsch-Plotkin radical of G has infinite‎ ‎rank or G is nilpotent-by-abelian (and has infinite rank)‎.