Supersolvable Groups Research Articles

Modules over Crossed Products

J. T. Stafford (1978,J. London Math. Soc. (2)18, 429–442) proved that any left ideal of the Weyl algebraAn(K) over a fieldKof characteristic 0 can be generated by two elements. In general, there is the problem of determining whether any left ideal of a Noetherian simple domain can be generated by two elements. In this work we show that this property holds for some crossed products of a simple ring with a supersolvable group and also for the tensor product of generalized Weyl algebras. We also prove that these rings are stably generated by 2 elements and that their finitely generated torsion left modules can be generated by two elements. Some results about stably 2-generated rings were found by V. A. Artamonov (1994,Math. Sb.185, No. 7, 3–12).

Finite groups with complementable maximal primary cyclic subgroups

We study finite nonprimary groups with complementable maximal primary cyclic subgroups and give a description of all supersolvable groups of this sort.

Computing irreducible representations of supersolvable groups over small finite fields

We present an algorithm to compute a full set of irreducible representations of a supersolvable group G G over a finite field K K , char ⁡ K ∤ | G | \operatorname {char} K\nmid |G| , which is not assumed to be a splitting field of G G . The main subroutines of our algorithm are a modification of the algorithm of Baum and Clausen (Math. Comp. 63 (1994), 351–359) to obtain information on algebraically conjugate representations, and an effective version of Speiser’s generalization of Hilbert’s Theorem 90 stating that H 1 ( Gal ⁡ ( L / K ) , GL ⁡ ( n , L ) ) H^{1}(\operatorname {Gal}(L/K), \operatorname {GL}(n,L)) vanishes for all n ≥ 1 n\ge 1 .

A Frobenius-type theorem for supersolvable groups

A Frobenius-type theorem for supersolvable groups

Torsion Units in Integral Group Rings

AbstractSpecial cases of Bovdi's conjecture are proved. In particular the conjecture is proved for supersolvable and Frobenius groups. We also prove that if is finite, α ∊ VℤG a torsion unit and m the smallest positive integer such that αm ∊ G then m divides .

Computing Irreducible Representations of Supersolvable Groups

Recently, it has been shown that the ordinary irreducible represen- tations of a supersolvable group G of order n given by a power-commutator presentation can be constructed in time 0(n2 log«). We present an improved algorithm with running time 0(n log/?).

Computing irreducible representations of supersolvable groups

Recently, it has been shown that the ordinary irreducible representations of a supersolvable group G of order n given by a power-commutator presentation can be constructed in time O ( n 2 log ⁡ n ) O({n^2}\log n) . We present an improved algorithm with running time O ( n log ⁡ n ) O(n\log n) .

GROUP RINGS WITH STRONGLY 2-GENERATED AUGMENTATION IDEALS

Suppose that $G$ is a finite supersolvable (infinite solvable) group, $K$ is a field of char $p>0$. Then the Augmentation ideal $w(K[G])$ is right strongly 2-generated iff $G$ is a $p'$-group-by-cyclic $P$-group (finite $p'$-group-by-infinite cyclic).

Some results on π‐solvable and supersolvable groups

For a finite group G, ϕp(G), Sp(G), L(G) and S𝒫(G) are generalizations of the Frattini subgroup of G. We obtain some results on π‐solvable, p‐solvable and supersolvable groups with the help of the structures of these subgroups.

Existence and efficient construction of fast Fourier transforms on supersolvable groups

The linear complexityL K(A) of a matrixA over a fieldK is defined as the minimal number of additions, subtractions and scalar multiplications sufficient to evaluateA at a generic input vector. IfG is a finite group andK a field containing a primitive exp(G)-th root of unity,L K(G):= min{L K(A)|A a Fourier transform forKG} is called theK-linear complexity ofG. We show that every supersolvable groupG has amonomial Fourier Transform adapted to a chief series ofG. The proof is constructive and gives rise to an efficient algorithm with running timeO(|G|2log|G|). Moreover, we prove that these Fourier transforms are efficient to evaluate:L K(G)≤8.5|G|log|G| for any supersolvable groupG andL K(G)≤1.5|G|log|G| for any 2-groupG.

Arithmetic automorphism group of simple cycles

It is shown that the set of all arithmetic automorphisms of a simple cycle forms a supersolvable perfect group with a new operation. The representation problem is solved for this group.

On Supersolvable Groups and a Theorem of Huppert

AbstractWe obtain the following generalization of a well known result of Huppert. If p is the largest primer divisor of the order of a finite group G and q is any prime distinct from p, then G is supersolvable if and only if every maximal subgroup whose index is relatively prime to either p or q, has prime index.

Locally supersolvable extensions of Abelian groups

An extension E of an Abelian group A by a locally supersolvable group is studied under the assumption that A has a finite series of subgroups normal in E, each of whose factors satisfies one of the following conditions: 1) min-E; 2) max-E; 3) the factor has finite rank and is torsion-free. It is proved that if E induces in the factors of the series in A hypercyclic groups of automorphisms and A is the locally supersolvable coradical of E, then A is complemented in E and all complements are conjugate in E. An analog of this result for locally nilpotent extensions of Abelian groups is also noted.

On the Intersection of a Class of Maximal Subgroups of a Finite Group

Let G be a finite group and Xt a set of primes. We consider the family of subgroups of G: _7 = {M: M <-G, [G: M]X = 1, [G: M] is composite} and denote Sx(G) = nf{M: M e _} if _ is non-empty, otherwise Sx(G) = G. The purpose of this note is to prove Theorem. Let G be a 7r-solvable group. Then S,,(G) has the following properties: (1) S, (G)/O (G) is supersolvable. (2) Sn(Sn(G)) = Sn(G). (3) G/On(G) is supersolvable if and only if S, (G) = G. There has been much interest in the past in considering various generalisations of the Frattini subgroup of a finite group and to investigate the influence of such a subgroup on the structure of the group (see, Deskins [1], Gaschiitz [2], Rose [3], and [4]). In [4], P. Bhattacharya and N. P. Mukherjee introduce a subgroup S,(G) and exhibit its relationship with the given group G. The objective of this paper is to investigate the subgroup S, (G) further. We obtain the following result. Theorem. Let G be a finite 7r-solvable group, then S, (G)/O (G) is a supersolvable group. Let 7r be any set of primes and 7r' the complementary set of primes. Let G be a finite group. Then we denote M < G to indicate that M is a maximal subgroup of G. Also, [G: M], denotes the 7-part of [G: M]. Now consider the following family of subgroups: 9 = {M: M <-G, [G: M], = 1, [G: M] is composite}. Definition. S,(G) = n{M: M E } if Y is non-empty, otherwise S,(G) = G. Clearly, S,(G) < G, ??(G) < S7(G) and O(G) < S,(G), where 4?(G) is the Frattini subgroup of the group G and O (G) is the maximal normal 7-subgroup of G. All groups in this paper are finite. Received by the editors September 22, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 20D20.

The cyclic length of ap-group need not be logarithmic

The cyclic length l(G) of a finite supersolvable group G is defined to be the smallest number l such that there is a series G = N o > N I > . . . > N ~ = I , where No, N1 . . . . . Nl are l+1 different normal subgroups of G and all factor groups Ni_l/Ni (i=1 . . . . . l) are cyclic. L. R6dei asked the question whether or not the cyclic lengh has a logarithmic property on the class of p-groups: i.e. does l(G1 • G2) = =l(G1)+l(G2) hold, whenever Gx, G2 are p-groups for a certain prime p? At the first glance the answer could be expected to be in the positive, especially in view of the validity on the class of abelian p-groups. In the present note we will bring this expectation to nought not only by means of a single counterexample but more generally by assigning a p-group G to each abelian p-group H so that l ( G • < l ( G ) + l ( H ) takes place. The notation will be standard and can be found in [1]. All groups under consideration are finite.

Fuzzy subgroups: Some characterizations. II

We consider an analysis of fuzzy subgroups in terms of the corresponding family of level subgroups. Given a finite chain of subgroups of a group G, we prove that there exists a fuzzy subgroup of G whose level subgroups are exactly the subgroups of this chain. As a corollary we obtain an interpretation of the number of chains of subgroups of a group G in which a subgroup H is a member. When the group G is a supersolvable group, some further interpretation of this number is obtained.

A NOTE ON FINITE GROUPS SATISFYING PERMUTIZER CONDITION

If G satisfies the permutizer condition, we call G a pc-group; clearly, all the super-solvable groups are pc-groups. It was shown in [1] that pc-groups of odd order were supersolvable. It is easy to verify that the symmetric group of degree four is a solvable pc-group, but it is not supersolvable. In general, solvable pc-groups

A theorem of Hall-Higman type for supersolvable groups

Let AC be a finite solvable group with GCIAG and (IAl, [Cl)= 1. Assume P E G is an extraspecial p-subgroup, for some prime p, PcIAG, Z(P) s Z(AG) and A acts faithfully on P/Z(P). Let k be an algebraically closed field of characteristic q #p and let M be a kAG-module such that Z(P) is not trivial on M. In their landmark paper [S], Hall and Higman state some conditions that insure that M(, contains a direct summand isomorphic to kA, in the case where A is cyclic of prime power order. Many authors have extended their results. Notably, Berger [ 11, building on some results of Dade [2], has given some sufficient conditions in the case where A is nilpotent. In [S] the case where char(k) /’ IAl and A is supersolvable is analyzed. In the present paper we look at the case where A is supersolvable and char(k) ) IAl. We prove (Theorem 3.2 below) that, if A does not involve certain groups-essentially wreath products and norm groups (see definitions l.l)-then Ml A contains a direct summand isomorphic to kA. As applications of this result we obtain, in Section 4, some results on modular linear groups with a normal Hall subgroup and a supersolvable complement. For example in Theorem 4.2 we obtain roughly the following result. Let A be a finite supersolvable group which does not involve certain groups. Suppose AC (with GQAG and (IA\, ICI)= 1) is a group of linear transformations of the k-vector space V with 1/l, homogeneous. Then if A acts fixed point freely on V there is some non-trivial normal subgroup of A which centralizes most of G and acts fixed point freely on V. In 1.1 and 4.1, the reader is provided with the necessary definitions for our results. The proof of Theorem 3.2, in the case where char(k) / IAJ, provides a new proof of a slightly weaker version of [S, Theorem 3.41 which does not depend on [S, Theorem 3.21. The results in Section 4 are similar to some nonmodular results in [S]. The reader is given precise directions as to how to adapt their proofs to get these new results. 194 0021-8693/85 $3.00

Supersolvable groups satisfying the minimality condition, all of whose primary subgroups are abelian

Supersolvable groups satisfying the minimality condition, all of whose primary subgroups are abelian

On the density of various classes of groups

Let T be a subset of the set of all isomorphism classes of finite groups. We consider the number F g ( x) of positive integers n≤ x such that all groups of order n lie in T . When T consists of the isomorphism classes of all finite groups of any of the following types, we obtain an asymptotic formula for F g ( x): cyclic groups, abelian groups, nilpotent groups, supersolvable groups, and solvable groups. In the course of the arguments, we also obtain, for almost all n, a lower bound for the number of groups of a given order n.