A characterization of PGL(2,q),q ODD

LetG be a finite group andA andB solvable subgroups ofG, such thatG=AB and 2 is the only common prime divisor ofA andB. Under suitable restrictions of the 2-structure ofA andB, it is shown that eitherG is solvable orG contains a solvable normal subgroupN, such thatG/N contains a normal subgroup, which is isomorphic to PGL(2,q),q odd.     

Nilpotent injectors in alternating groups

AnN-Injector in an arbitrary finite group is defined as a maximal nilpotent subgroup ofG containing a subgroupA ofG of maximal order, satisfying class (A)≦2. In a previous paper theN-Injectors of Sym(n) were determined. In this paper we determine theN-Injectors of Alt(n), after having determined the set of all nilpotent subgroups,A, of Sym(n) of maximal order satisfying class(A)≦2. It is shown that the set ofN-Injectors of Alt(n) consists of a unique conjugacy class, and ifn≠9, it coincides with the set of the nilpotent subgroups of Alt(n) of maximal order.     

Trace polynomials and invariant theory

LetA be a finite-dimensional simple (non-associative) algebra over an algebraically closed fieldF of characteristic 0. LetG be the group of its automorphisms which acts onkA, the direct sum ofk copies ofA. SupposeA is generated byk elements. In this paper, generators of the field of rational invariantF(kA) ^G are described in terms of operations of the algebraA.     

Nilpotent injectors in symmetric groups

AnN-Injector in an arbitrary finite groupG is defined as a maximal nilpotent subgroup ofG, containing a subgroupA ofG of maximal order satisfying class(A)>=2. Among other results theN-Injectors of Sym(n) are determined and shown to consist of a unique conjugacy class of subgroups of Sym(n).     

The residual nilpotency of the augmentation ideal and the residual nilpotency of some classes of groups

The following theorem is proved: LetG be any group. Then the augmentation ideal ofZG is residually nilpotent if and only ifG is approximated by nilpotent groups without torsion or discriminated by nilpotent p_i,-groups,i ∈I, of finite exponents. This theorem is applied to obtain conditions under which the groupsF/N′ are residually nilpotent whereF is a free non-cyclic group and N?F.     

Group algebras with almost maximal lie nilpotency index

LetK G be a non-commutative Lie nilpotent group algebra of a groupG over a fieldK. It is known that the Lie nilpotency index ofKG is at most |G′|+1, where |G′| is the order of the commutator subgroup ofG. In [4] the groupsG for which this index is maximal were determined. Here we list theG’s for which it assumes the next highest possible value. The present paper is a part of the PhD dissertation of the author.     

Finite soluble groups containing an element of prime order whose centralizer is small

In [2] we proved that ifG is a finite group containing an involution whose centralizer has order bounded by some numberm, thenG contains a nilpotent subgroup of class at most two and index bounded in terms ofm. One of the steps in the proof of that result was to show that ifG is soluble, then ¦G/F(G) ¦ is bounded by a function ofm, where F (G) is the Fitting subgroup ofG. We now show that, in this part of the argument, the involution can be replaced by an arbitrary element of prime order.     

Products of groups and group classes

Letχ be a Schunck class, and let the finite groupG=AB=BC=AC be the product of two nilpotent subgroupsA andB andχ-subgroupC. If for every common prime divisorp of the orders ofA andB the cyclic group of orderp is anχ-group, thenG is anχ-group. This generalizes earlier results of O. Kegel and F. Peterson. Some related results for groups of the formG=AB=AK=BK, whereK is a nilpotent normal subgroup ofG andA andB areχ-groups for some saturated formationχ, are also proved.     

Torsion-free groups with factor-groups on their hypercenter which are periodic

Assume that G is a torsion-free group, Zk(G) is the k-th term of the upper central series of G, and ¯G_k=G/Z_k(G) is a nontrivial periodic group. Then every finite subgroup of ¯G_k is nilpotent of class not higher than k; the group k 2 contains an infinite subgroup with k generators if k2 and two generators if k=1. Moreover any nontrivial invariant subgroup of ¯G_k is infinite. All elements of ¯Gk are of odd order. This assertion is generalized.Translated from Matematicheskie Zametki, Vol. 8, No. 3, pp. 373–383, September, 1970.     

Odd factors of a graph

LetG be a graph and letf be a function defined on V(G) such that f(x) is a positive odd integer for everyx ? V(G). A spanning subgraphF ofG is called a [l,f]-odd factor of G if d_F(x) ? {1,3,2026, f(x)} for every x ?V(G), whered _F (x) denotes the degree of x inF. We discuss several conditions for a graphG to have a [1,f]-odd factor.     

Products of F*(<Emphasis Type="Italic">G</Emphasis>)-subnormal subgroups of finite groups

A subgroup H of a finite group G is called F*(G)-subnormal if H is subnormal in HF*(G). We show that if a group Gis a product of two F*(G)-subnormal quasinilpotent subgroups, then G is quasinilpotent. We also study groups G = AB, where A is a nilpotent F*(G)-subnormal subgroup and B is a F*(G)-subnormal supersoluble subgroup. Particularly, we show that such groups G are soluble.     

A new characterization of frobenius complements

LetH, G be finite groups such thatH acts onG and each non-trivial element ofH fixes at mostf elements ofG. It is shown that, ifG is sufficiently large, thenH has the structure of a Frobenius complement. This result depends on the classification of finite simple groups. We conclude that, ifG is a finite group andA ⊆G is any non-cyclic abelian subgroup, then the order ofG is bounded above in terms of the maximal order of a centralizerC _G(a) for 1≠a ∈A.     

On Lie algebra decompositions related to spherical homogeneous spaces

LetG be a connected, reductive, linear algebraic group over an algebraically closed fieldk of characteristik zero. LetH ₁ andH ₂ be two spherical subgroups ofG. It is shown that for allg in a Zariski open subset ofG one has a Lie algebra decomposition g = h₁ + Adg ? h₂, where a is the Lie algebra of a torus and dim a ≤ min (rankG/H ₁,rankG/H ₂). As an application one obtains an estimate of the transcendence degree of the fieldk(G/H ₁ xG/H ₂) ^G for the diagonal action ofG. Ifk = ? andG _a is a real form ofG defined by an antiholomorphic involution σ :G→G then for a spherical subgroup H ? G and for allg in a Hausdorff open subset ofG one has a decomposition g = g_a + a Adg ? h, where a is the Lie algebra of σ-invariant torus and dim a ≤ rankG/H.     

Reduction theory for a rational function field

LetG be a split reductive group over a finite field F_q. LetF = F_q(t) and let A denote the adèles ofF. We show that every double coset inG(F)/G(A)/K has a representative in a maximal split torus ofG. HereK is the set of integral adèlic points ofG. WhenG ranges over general linear groups this is equivalent to the assertion that any algebraic vector bundle over the projective line is isomorphic to a direct sum of line bundles.     

Traces of Eichler—Brandt matrices and type numbers of quaternion orders

LetA be a totally definite quaternion algebra over a totally real algebraic number fieldF andM be the ring of algebraic integers ofF. For anyM-orderG ofA we derive formulas for the massm(G) and the type numbert(G) of G and for the trace of the Eichler-Brandt matrixB(G, J) ofG and any integral idealJ ofM in terms of genus invariants ofG and of invariants ofF andJ. Applications to class numbers of quaternion orders and of ternary quadratic forms are indicated.     

Finite arithmetic subgroups ofGL n,III

LetG be an algebraic group inGL _n (C) defined over Q, andK an algebraic number field with the maximal orderO _k . If the groupG(O _k ) of rational points ofG inM _n (O _k ) is a finite group and if it satisfies a certain condition, which is satisfied, for example, whenK is a nilpotent extension of Q and 2 is unramified, thenG(O _k ) is generated by roots of unity inK andG(Z). Dedicated to the memory of Professor K G Ramanathan     

An intersection theorem for multivalued maps

LetE be a Banach space and letV, W ? E be two closed subspaces such thatE=V ⊕ W. LetF, G: E-· E be two multivalued maps. The problem we are concerned with is to give conditions onF andG ensuring the nonemptiness ofF(V) ∩ G(W).     

On primary ideals in group algebras

LetG be a -compact locally compact group such thatG/Z(G), whereZ(G) denotes the center ofG, has a relatively compact commutator subgroup. It is shown that primary ideals inL ¹(G) are maximal.     

Quasinormalität in topologischen Gruppen

LetQ be a subgroup of the locally compact groupG. Q is called a topologically quasinormal subgroup ofG, ifQ is closed and for each closed subgroupA ofG. We prove: If the compact elements ofG form a proper subgroup, compact topologically quasinormal subgroups ofG are subnormal of defect 2. IfG is connected, compact topologically quasinormal subgroups ofG are normal. IfG/G ₀ is compact, connected topologically quasinormal subgroups ofG are normal.     

Multiplicity formulas for finite dimensional and generalized principal series representations

The article presents two results. (1) Let a be a reductive Lie algebra over ℂ and let b be a reductive subalgebra of a. The first result gives the formula for multiplicity with which a finite dimensional irreducible representation of b appears in a given finite dimensional irreducible representation of a in a general situation. This generalizes a known theorem due to Kostant in a special case. (2) LetG be a connected real semisimple Lie group andK a maximal compact subgroup ofG. The second result is a formula for multiplicity with which an irreducible representation ofK occurs in a generalized representation ofG arising not necessarily from fundamental Cartan subgroup ofG. This generalizes a result due to Enright and Wallach in a fundamental case.