Automorphisms of a Class of Finite p-groups with a Cyclic Derived Subgroup

AbstractLetGbe a finite group. In this paper we obtain some sufficient conditions for the supersolubility ofGwith two supersoluble non-conjugate subgroupsHandKof prime index, not necessarily distinct. It is established that the supersoluble residual of such a group coincides with the nilpotent residual of the derived subgroup. We prove thatGis supersoluble in the following cases: one of the subgroupsHorKis nilpotent; the derived subgroup{G^{\prime}}ofGis nilpotent;{|G:H|=q>r=|G:K|}andHis normal inG. Also the supersolubility ofGwith two non-conjugate maximal subgroupsMandVis obtained in the following cases: all Sylow subgroups ofMand ofVare seminormal inG; all maximal subgroups ofMand ofVare seminormal inG.

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On a Theorem Of Baer and Higman

Canadian Journal of Mathematics ◽

10.4153/cjm-1956-031-x ◽

1956 ◽

Vol 8 ◽

pp. 263-270 ◽

Cited By ~ 7

Author(s):

Seân Tobin

Keyword(s):

Finite Number ◽

Common Multiple ◽

Least Common Multiple ◽

Derived Subgroup

1.1 Baer has shown (1) that if the fact that the exponent of a group is m (that is, m is the least common multiple of the periods of the elements) implies a limitation on the class of the group, then m must be a prime. Graham Higman has extended this result by proving (3) that for any given integer M there are at most a finite number of prime powers q other than primes, such that the fact that a group has exponent q implies a limitation on the class of the Mth derived subgroup.

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Coprime Group Actions Fixing All Nonlinear Irreducible Characters

Canadian Journal of Mathematics ◽

10.4153/cjm-1989-003-2 ◽

1989 ◽

Vol 41 (1) ◽

pp. 68-82 ◽

Cited By ~ 34

Author(s):

I. M. Isaacs

Keyword(s):

Finite Groups ◽

Irreducible Character ◽

Group Actions ◽

Character Degrees ◽

Prime Divisors ◽

Irreducible Characters ◽

Nilpotent Length ◽

Derived Subgroup ◽

Nonlinear Irreducible Character

The main result of this paper is the following:Theorem A. Let H and N be finite groups with coprime orders andsuppose that H acts nontrivially on N via automorphisms. Assume that Hfixes every nonlinear irreducible character of N. Then the derived subgroup ofN is nilpotent and so N is solvable of nilpotent length≦ 2.Why might one be interested in a situation like this? There has been considerable interest in the question of what one can deduce about a group Gfrom a knowledge of the setcd(G) = ﹛x(l)lx ∈ Irr(G) ﹜of irreducible character degrees of G.Recently, attention has been focused on the prime divisors of the elements of cd(G). For instance, in [9], O. Manz and R. Staszewski consider π-separable groups (for some set π of primes) with the property that every element of cd(G) is either a 77-number or a π'-number.

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Groups with many subgroups having polycyclic-by-finite layers or derived subgroup

Communications in Algebra ◽

10.1080/00927872.2017.1347659 ◽

2017 ◽

Vol 46 (3) ◽

pp. 1344-1352

Author(s):

Aziza Rezig ◽

Nadir Trabelsi

Keyword(s):

Derived Subgroup

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Commutator Length of Finitely Generated Linear Groups

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/2008/281734 ◽

2008 ◽

Vol 2008 ◽

pp. 1-5

Author(s):

Mahboubeh Alizadeh Sanati

Keyword(s):

Finite Group ◽

Upper Bound ◽

Quadratic Polynomial ◽

Linear Groups ◽

Number Of Generators ◽

Derived Subgroup ◽

Finite Linear Group ◽

Minimum Number ◽

Commutator Length ◽

Finite Linear

The commutator length “” of a group is the least natural number such that every element of the derived subgroup of is a product of commutators. We give an upper bound for when is a -generator nilpotent-by-abelian-by-finite group. Then, we give an upper bound for the commutator length of a soluble-by-finite linear group over that depends only on and the degree of linearity. For such a group , we prove that is less than , where is the minimum number of generators of (upper) triangular subgroup of and is a quadratic polynomial in . Finally we show that if is a soluble-by-finite group of Prüffer rank then , where is a quadratic polynomial in .

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Categorizing finite p-groups by the order of their non-abelian tensor squares

Journal of Algebra and Its Applications ◽

10.1142/s021949881650095x ◽

2016 ◽

Vol 15 (05) ◽

pp. 1650095 ◽

Cited By ~ 1

Author(s):

S. Hadi Jafari

Keyword(s):

Upper Bound ◽

Derived Subgroup

Let [Formula: see text] be a non-abelian [Formula: see text]-generator finite [Formula: see text]-group of order [Formula: see text]. Ellis and McDermott in 1996 proved that [Formula: see text]. In the present paper, we improve this upper bound and show that [Formula: see text]. Also the [Formula: see text]-groups with derived subgroup of order [Formula: see text] which attain the bound are obtained. Among other results, we classify all finite [Formula: see text]-groups of order [Formula: see text], for [Formula: see text], with [Formula: see text], where [Formula: see text].

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