Free subgroups in maximal subgroups of SLn(D)

Given a non-commutative finite-dimensional [Formula: see text]-central division ring [Formula: see text], [Formula: see text] a subnormal subgroup of [Formula: see text] and [Formula: see text] a non-abelian maximal subgroup of [Formula: see text], then either [Formula: see text] contains a non-cyclic free subgroup or there exists a non-central maximal normal abelian subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] is a subfield of [Formula: see text], [Formula: see text] is Galois and [Formula: see text], also [Formula: see text] is a finite simple group with [Formula: see text].

On gauging finite subgroups

We study in general spacetime dimension the symmetry of the theory obtained by gauging a non-anomalous finite normal Abelian subgroup AA of a \GammaΓ-symmetric theory. Depending on how anomalous \GammaΓ is, we find that the symmetry of the gauged theory can be i) a direct product of G=\Gamma/AG=Γ/A and a higher-form symmetry \hat AÂ with a mixed anomaly, where \hat AÂ is the Pontryagin dual of AA; ii) an extension of the ordinary symmetry group GG by the higher-form symmetry \hat AÂ; iii) or even more esoteric types of symmetries which are no longer groups. We also discuss the relations to the effect called the H^3(G,\hat A)H3(G,Â) symmetry localization obstruction in the condensed-matter theory and to some of the constructions in the works of Kapustin-Thorngren and Wang-Wen-Witten.

On some polycyclic groups with small Hirsch length II

A polycyclic group [Formula: see text] is called a [Formula: see text]-group ([Formula: see text]-group) if every normal abelian subgroup (abelian subgroup) of any finite quotient of [Formula: see text] is generated by [Formula: see text], or fewer, elements and [Formula: see text] is the least integer with this property. In this paper, we describe the structures of [Formula: see text]-groups and [Formula: see text]-groups, and bound the number of generators of [Formula: see text]-groups and the derived lengths of [Formula: see text]-groups, which is a continuation of [H. G. Liu, F. Zhou and T. Xu, On some polycyclic groups with small Hirsch length, J. Algebra Appl. 16(11) (2017) 17502371–175023710].

Finite Groups in Which the Automizers of Some Abelian Subgroups are Trivial

If H is a subgroup of a finite group G, then the automizer AutG(H) of H in G is defined as the group of automorphisms of H induced by conjugation by elements of NG(H). A finite group G is called an NNC-group if for any non-normal abelian subgroup A, either [Formula: see text] or [Formula: see text]. In this paper, classifications of nilpotent NNC-groups and non-solvable NNC-groups are given. We also investigate the solvable NNC-groups and describe the structure of solvable NNC-groups.

On some polycyclic groups with small Hirsch length

A polycyclic group [Formula: see text] is called an [Formula: see text]-group if every normal abelian subgroup of any finite quotient of [Formula: see text] is generated by [Formula: see text], or fewer, elements and [Formula: see text] is the least integer with this property. In this paper, the structure of [Formula: see text]-groups and [Formula: see text]-groups is determined.

On the number of minimal nonabelian subgroups in a nonabelian finite p-group

The number of minimal nonabelian subgroups in a nonabelian finite [Formula: see text]-group containing a maximal normal abelian subgroup of given index is estimated. Also the [Formula: see text]-group for which the above estimate is attained, are described in great detail.

Augmentation quotients and dimension subgroups of semidirect products

Sandling(6) determined the dimension subgroups of the semidirect product of a normal abelian subgroup and a subgroup; namely if G = NT is the semidirect product of a normal abelian subgroup N and a subgroup T, then the mth dimension subgroup Dm(G) of G is equal to [N, (m – 1) G] · Dm (T) for all m ≧ 1, where

Abelian p-subgroups of the general linear group

A. J. Weir [1] has found the maximal normal abelian subgroups of the Sylow p-subgroups of the general linear group over a finite field of characteristic p, and a theorem of J. L. Alperin [2] shows that the Sylow p-subgroups of the general linear group over finite fields of characteristic different from p have a unique largest normal abelian subgroup and that no other abelian subgroup has order as great.

On the Semisimplicity of Modular Group Algebras. II

Let G be a discrete group, let Kbe an algebraically closed field of characteristic p > 0 and let KGdenote the group algebra of Gover K.In a previous paper (2) I studied the Jacobson radical JKGof KGfor groups Gwith big abelian subgroups or quotient groups. It is therefore natural to next consider metabelian groups, and I do this here. The main result is as follows.THEOREM 1. Let K be an algebraically closed field of characteristic p and let a group G have a normal abelian subgroup A with G/A abelian. Then JKG ≠ {0} if and only if G has an element g of order p such that the A-conjugacy class gA is finite and such that the group is periodic.Note that since and G/Ais abelian, we do in fact have .

On the Isomorphism of Integral Group Rings. II

Let Z(G) denote the integral group ring of a group G. Let be the class of groups G with the property that for any isomorphism θ: Z(G) → Z(H), we have θ(g) = ±h, h ∈ H, for all g ∈ G. We study this class in § 2 and establish that it contains classes of torsion-free abelian groups, torsion abelian groups, and ordered groups.In § 4, we prove the following result.THEOREM. Let G be a group which contains a normal abelian subgroup A such that. Suppose that θ: Z(G) → Z(H) is an isomorphism such that θ(Δ(G, A)) = Δ(H, B) for a suitable normal subgroup B of H. Then G ≃ H. (Here Δ(G, A) is the kernel of the natural map Z(G) → Z(G/A).)Jackson (3) and Whitcomb (6) proved the special case of this theorem when G is supposed to be finite metabelian. The lemmas needed are given in §3.