Let [Formula: see text] be a class of finite groups closed under taking subgroups, homomorphic images and extensions. It is known that if normal subgroup group then the image an text]-maximal in not, general, text]. We say reduction text]-theorem holds for if, every extension (i.e. contains as subgroup), number conjugacy classes subgroups same. The implies In this paper, we prove only all are c...

a subgroup $h$ is said to be $nc$-supplemented in a group $g$ if there exists a subgroup $kleq g$ such that $hklhd g$ and $hcap k$ is contained in $h_{g}$, the core of $h$ in $g$. we characterize the supersolubility of finite groups $g$ with that every maximal subgroup of the sylow subgroups is $nc$-supplemented in $g$.

The maximality of certain symplectic subgroups of unitary groups PSUn(K), n ≥ 4, (K any field admitting a non–trivial involutory automorphism) belonging to the class C5 of Aschbacher is proved. Furthermore some related geometry in the case n = 4 and K finite is investigated. Mathematics Subject Classification (2002): 20G40, 20G28

Let [p] = {1, . . . , p} denote a totally ordered alphabet on p letters, and let α = (α1, . . . ,αm) ∈ [p1], β = (β1, . . . ,βm) ∈ [p2]. We say that α is order-isomorphic to β if for all 1≤ i< j ≤ m one has αi < α j if and only if βi < β j. For two permutations π ∈ Sn and τ ∈ Sk, an occurrence of τ in π is a subsequence 1 ≤ i1 < i2 < .. . < ik ≤ n such that (πi1 , . . . ,πik ) is order-isomorph...

THEOREM. Let F be the free profinite group on a set X, where \X\ > 2, and let n be a non-empty set of primes. Then F has a maximal abelian subgroup isomorphic to HpEn Zp. The idea of the proof is the following: we show that A — Ylpe7I1p is a free factor of Pa, i.e. fia ^ A *B for some profinite group B. To conclude from this that A is a maximal abelian subgroup of Fa (the general case then foll...

The elliptic elements of M, each with two conjugate complex fixed points, are precisely the conjugates of nontrivial powers of A and B. The parabolic elements, each with a single real fixed point, are precisely the conjugates of nontrivial powers of C = AB: z ~ z + 1. The remaining nontrivial elements of M are hyperbolic, each with two real fixed points. A subgroup S of M is torsionfree (and th...

In 1] M. Aschbacher described the structure of the possible maximal subgroups of a classical group G with natural module M. Each such subgroup is either a member of a canonical class of subgroups, or it is the normalizer of a quasi-simple, absolutely irreducible subgroup H of G. In the latter case suppose that H is a classical group whose deening characteristic is coprime to that of G. The aim ...

we call $h$ an $ss$-embedded subgroup of $g$ if there exists a‎ ‎normal subgroup $t$ of $g$ such that $ht$ is subnormal in $g$ and‎ ‎$hcap tleq h_{sg}$‎, ‎where $h_{sg}$ is the maximal $s$-permutable‎ ‎subgroup of $g$ contained in $h$‎. ‎in this paper‎, ‎we investigate the‎ ‎influence of some $ss$-embedded subgroups on the structure of a‎ ‎finite group $g$‎. ‎some new results were obtained.‎