Frobenius Subgroups Research Articles

On Groups with a Frobenius Element

We study groups with an H-Frobenius element and the nilpotent kernels of the corresponding Frobenius subgroups. We prove the two theorems that solve Question 10.61 of The Kourovka Notebook in this case.

On groups with Frobenius elements

We find the conditions under which the set-theoretic union of the kernels of two-generated Frobenius subgroups of a group G with fixed cyclic complement of order 3n is a normal subgroup in G.

Hopf algebras, distributive (Laplace) pairings and hash products: a unified approach to tensor product decompositions of group characters

We show for bicommutative graded connected Hopf algebras that a certain distributive (Laplace) subgroup of the convolution monoid of 2-cochains parameterizes certain well behaved Hopf algebra deformations. Using the Laplace group, or its Frobenius subgroup, we define higher derived hash products, and develop a general theory to study their main properties. Applying our results to the (universal) bicommutative graded connected Hopf algebra of symmetric functions, we show that classical tensor product and character decompositions, such as those for the general linear group, mixed co- and contravariant or rational characters, orthogonal and symplectic group characters, Thibon and reduced symmetric group characters, are special cases of higher derived hash products. In the appendix we discuss a relation to formal group laws.

A Nonsimplicity criterion for groups

Nonsimplicity criteria for groups with systems of Frobenius subgroups are crucial in research on infinite groups with finite elements. Here we prove a criterion for nonsimplicity of a group with a system of F-subgroups whose properties are close to those of Frobenius groups.

Frobenius subgroups of free profinite products

We solve an open problem of Herfort and Ribes: profinite Frobenius groups of certain type do occur as closed subgroups of free profinite products of two profinite groups. This also solves a question of Pop about prosolvable subgroups of free profinite products.

Derived Functors Related to Wall-Crossing

The setting is the representation theory of a simply connected, semisimple algebraic group over a field of positive characteristic. There is a natural transformation from the wall-crossing functor to the identity functor. The kernel of this transformation is a left exact functor. This functor and its first derived functor are evaluated on the global sections of a line bundle on the flag variety. It is conjectured that the derived functors of order greater than one annihilate the global sections. Also, the principal indecomposable modules for the Frobenius subgroups are shown to be acyclic.

A note on a conjecture of K. Harada and strongly p-embedded Frobenius subgroups

For a p-block B of a finite group G, it is well known that Σχi ∈ Bχi(1)χi vanishes on all p-singular elements of G. The converse proposition was proposed by K. Harada and partially proved by several authors using decomposition matrices. We give a partial answer without using decomposition matrices in the case where G has a strongly p-embedded subgroup.

Families of solvable Frobenius subgroups in finite groups

We introduce the notion of abelian system on a finite group G, as a particular case of the recently defined notion of kernel system (see this Journal, September 2001). Using a famous result of Suzuki on CN-groups, we determine all finite groups with abelian systems. Except for some degenerate cases, they turn out to be special linear group of rank 2 over fields of characteristic 2 or Suzuki groups. Our ideas were heavily influenced by [1] and [8].

New symmetric designs and nonabelian difference sets with parameters(100, 45, 20)

Six nonisomorphic new symmetric designs with parameters (100, 45, 20) are constructed by action of the Frobenius group E25 · Z12. This group proves to be their full automorphism group. Its Frobenius subgroup of order 100 acts on the designs as their nonabelian Singer group. The result is presented through six nonisomorphic new nonabelian (100, 45, 20) difference sets as well. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 291–299, 2000

Oval configurations of involutions in symmetric groups

We show that there are no oval configurations of involutions in the symmetric group S14 invariant under conjugation by a Frobenius subgroup of order 39.

Loewy series of certain indecomposable modules for Frobenius subgroups

We imitate some approaches in infinite dimensional representation theory of complex semisimple Lie algebras by using the truncated category method in the categories of modules for certain Frobenius subgroups of a semisimple algebraic group over an algebraically closed field of characteristic p > 0 p > 0 . By studying the translation functors from p p -singular weights to p p -regular weights, we obtain some results on Loewy series of certain indecomposable modules.

Character formulas and Frobenius subgroups of algebraic groups

Let G be a connected semisimple algebraic group over a field of non-zero characteristic p, with a maximal torus T. On G we have the Frobenius endomorphism F induced by the pth power map on the underlying field. The (scheme-theoretic) kernel of F is the first Frobenius kernel G,, and the inverse image of T under F is the “Frobenius subgroup” G, T. The purpose of this paper is to study certain questions in the (rational) representation theory of G, T. Let B be a Bore1 subgroup of G which contains T. In [6], Cline, Parshall, and Scott obtained a result which describes the functor “induction from B to G” as the composite of functors of the form “induction from B to a minimal parabolic subgroup P” over a sequence of minimal parabolics coming from a given reduced expression of the long word in the Weyl group. In Section 5, we obtain the analogous result for G, T. In Section 6, we develop a theory of intertwines for a certain class of G, T-modules. This gives a direct proof of an extended strong linkage principle for G, T, independent of the strong linkage principle for G. (Originally, Jantzen [17] proved that strong linkage for G, T, and more generally for G, T, n > 1, follows from the strong linkage principle for G. 1 Recently we have applied this method to give a new proof of an extended strong linkage principle for G; see [ 121. From the results of Section 5 we can write down a formula for the image of one of the intertwines in our class of modules. (This also follows from results of Jantzen [16, 171.) Composing the intertwines in question, relative to the class of modules corresponding to a given character of T, gives a map whose image is the irreducible G, T-module of that highest weight. In case the character is a restricted dominant character, the image coincides with the irreducible G-module of that highest weight, by a result

Frobenius subgroups of free products of prosolvable groups

In this paper we establish the existence of profinite Frobenius subgroups in a free prosolvable productA ∐B of two finite groupsA andB. In this way the classification of solvable subgroups of free profinite groups is completed.

FINITE GROUPS WITH A FROBENIUS SUBGROUP

Suppose denotes a -subgroup of order of a group which is different from its normalizer in . A criterion for the simplicity of a group is obtained which includes the theorems of Feit and Ito on Zassenhaus groups of even degree and which is used to prove the followingTheorem. If and the order of the centralizer of each nonidentity element of in is odd, then . It is proved that if has a complement in and if does not divide , then has a nilpotent invariant complement in , and if is complemented by a Frobenius subgroup in the simple group , then , . Related to the results of Brauer, Leonard, and Sibley on finite linear groups is the followingTheorem. If the degree of each irreducible constituent of some faithful complex character of is less than , then either or , . Other results connected with the above theorems are also obtained.Bibliography: 24 titles.

Infinite groups saturated with Frobenius subgroups. II

Infinite groups saturated with Frobenius subgroups. II

Infinite groups saturated by frobenius subgroups

Infinite groups saturated by frobenius subgroups

Finite groups with Frobenius subgroup

Suppose the normalizer N of a subgroup A of a simple group G is a Frobenius group with kernel A, and the intersection of A with any other conjugate subgroup of G is trivial, and suppose, if A is elementary Abelian, that ¦a¦> 2n+1, where n=¦N:A¦. It is proved that if A has a complement B in G, then G acts doubly transitively on the set of right cosets of G modulo B, the subgroup B is maximal in G, and ¦B¦ is divisible by ¦a¦−1. The proof makes essential use of the coherence of a certain set of irreducible characters of N.

On finite groups which contain a frobenius subgroup

On finite groups which contain a frobenius subgroup

A remark on finite groups

If G is a finite group and H is a subgroup of G then we say that H is a Frobenius subgroup of G if xHx-'nH5 (1) if and only if xeH. Concerning such Frobenius subgroups there is a celebrated theorem due to Frobenius [1] which asserts: let T= {aEGjaEfxHx-1 all xEG or a= 1 }; then T is a normal subgroup of G. We call it the complementary subgroup of H. Suppose now that a finite group G has an automorphism cb leaving only 1 fixed. Thus the elements x-'b(x) are all distinct as x ranges over G, so they must fill out all of G. Let p be a prime which divides the order of G and let Sp be a p-Sylow subgroup of G. Then ?)(Sp) is also a p-Sylow subgroup of G and so cb(S,) =ySpyfor some yEEG. Pick xCG so that y-1 =x-l(x). Then as can immediately be verified, c(xSx-1) =xSx-1. That is, for each prime p there is a p-Sylow subgroup Sp, say, of G left set-wise invariant by c. We claim that Sp is unique in this regard. For if N(Sp) = { x G j xSx-1 = S, } then it is clear that ?)(N(Sp)) = N(Sp). Since N(Sp) is invariant under Xb, c induces an automorphism on N(Sp) leaving only 1 fixed, whence every element in N(Sp) can be written in the form n-10(n) with nCN(Sp). Thus if a-1l(a) CN(Sp), then a-lc(a) =n-lb(n) for some nEN and so a = n from which, of course, we have that aEN. Now suppose cb(Sp) = S, and ?)(Sp') =Sp' for some other p-Sylow subgroup. Since Sp' = aSpa-1 for some a E G, aSpa-1 = Sp' = ?)(Sp) = q5(aSpa-') =-c(a)Spq5(a)-1, leading to a-10(a) EN(Sp), and so aCN(Sp) and finally to Sp' = aSpa-1 =Sp. Thus Sp is indeed unique with respect to being left invariant by 4. If Vf is an automorphism of G which commutes with cb then since cb(Sp) = Sp, ciI'(Sp) =+ iP(Sp) = f (Sp), and so, since t'(Sp) is a p-Sylow subgroup left fixed by Xb, Vf(Sp) = Sp. Thus if W is an Abelian group of