Normal Sylow Research Articles

Construction of Finite Groups

We1999 Academic Pressintroduce three practical algorithms to construct certain finite groups up to isomorphism. The first one can be used to construct all soluble groups of a given order. This method can be restricted to compute the soluble groups with certain properties such as nilpotent, non-nilpotent or supersoluble groups. The second algorithm can be used to determine the groups of orderCopyright pn· qwith a normal Sylow subgroup for distinct primespandq. The third method is a general method to construct finite groups which we use to compute insoluble groups.

On the intersection of a certain class of maximal subgroups of a finite group

The authors prove that in a finite not p-solvable group the intersection Hp (G) of non p-nilpotent maximal subgroups (for a fixed odd prime p) is nilpotent. If G is p-solvable and \( \left (|G| ,{\overline {r}_p}(G)\right ) =1 \) then Hp (G) is either p-nilpotent or it has a normal Sylow p-subgroup.

Character degrees, derived length and Sylow normalizers

Let P be a Sylow p-subgroup of a monomial group G. We prove that dl $ ({\Bbb N}_G (P)/P') $ is bounded by the number of irreducible character degrees of G which are not divisible by p.

On groups and fuzzy subgroups

Abstract In this short communication, it is pointed out that the characterizations of abelian groups and normal Sylow subgroups in terms of fuzzy subgroups in Asaad and Abou-Zaid (1993) can be proved simply.

A classification of the stable type of 𝐵𝐺

We give a classification of the p-local stable homotopy type of BG, where G is a finite group, in purely algebraic terms. BG is determined by conjugacy classes of homomorphisms from p-groups into G. This classification greatly simplifies if G has a normal Sylow p-subgroup; the stable homotopy types then depends only on the Weyl group of the Sylow p-subgroup. If G is cyclic mod p {\bmod \;p} then BG determines G up to isomorphism. The last class of groups is important because in an appropriate Grothendieck group BG can be written as a unique linear combination of BH’s, where H is cyclic mod p {\bmod \;p} .

Finding Sylow normalizers in polynomial time

Given a set Γ of permutations of an n-set, let G be the group of permutations generated by Γ. If p is any prime, it is known that a Sylow p-subgroup P of G can be found in polynomial time. We show that the normalizer of P can also be found in polynomial time. In particular, given two Sylow p-subgroups of G, all elements conjugating one to the other can be found (as a coset of the normalizer of one of the Sylow p-subgroups). Analogous results are obtained in the case of Hall subgroups of solvable groups.

The λstructure of the green ring of GL(2, (Fp) in characteristic P, III

In this paper the reader is assumed to have taken notice of [I]. In [III] 1 we described the $lambda;, and s-, structure of the Green ring of GL(2,F p), and Sl(2,F p). We shall now construct a subring of the Green ring which is invariant for the $lambda;, and s-, operations. It is generated by all the indecomposables with odd-dimensional composition factors. This sheds another light on the results in the previous sections. We shall also study a certain quotient of the Green ring, which is in fact the Green ring of a certain subgroup of GL(2,F p) consisting of upper triangular matrices. The multiplication and the λ, s-, structure of this quotient Green ring is described. Moreover it is shown how this λ and s-, structure controls the deviation from being a λ, respectively s-, ring of the Green ring of any finite group with a normal Sylow subgroup of order p. The sequence of Adams operations for these groups is shown to be periodic, and the period reflects the internal p-structure of these groups.

Symmetric and general linear groups with supersoluble Sylow normalizers

Nel presente lavoro vengono caratterizzati i gruppi finiti che hanno i normalizzanti dei sottogruppi di Sylow supersolubili e che appartengono alle seguenti classi: gruppi simmetrici, alterni, lineari generali e lineari speciali.

Extensions of lattices overp-solvable groups

By PETER SCHMID Suppose G is a finite p-solvable group for some prime p. Let K be a finite extension of the p- adic number field 9 and k = R/r~R its residue class field. Let U, Vbe RG-lattices ( = R-free RG-modules of finite R-rank). The objective of this note is to record the following. Theorem. Ext~o (U, V) = 0 if either (i) K|174 is irreducible belonging to a block with cyclic defect group and K is unramified over •, or (ii) k| = k| is irreducible and the ramification index of K over 9 is less than p- 1 (and hence p odd). Statement (i) generalizes a result by Berman [11 who considered the case where G has a normal cyclic Sylow p-subgroup (and K = 9 It has been already noticed by Gudivok [31 that the result does not hold, in general, if K is ramified over 9 In a certain sense, both statements rely on the Fong-Swan theorem. In the first case we use the theory of blocks with cyclic defect groups and the fact that the Brauer tree is a star if G is p-solvable. The second statement follows from the uniqueness part of the lifting theorem in [61. Thus solvability is a crucial assumption. 1. Extensions of lattices. Let G be a finite group and let U, Vbe RG-lattices. Multiplication with 7r gives the exact sequence O-*V&V-,fz~O where f'= V/IrV~k| Since U is R- projective, the induced sequence 0--Hom R (U, V)&HomR (U, V)~Hom R (U, I7") ~0 is exact as well, and Extlo (U, V) = H 1 (G, Hom R (U, V)). Lemma. Ext~o(O,V)=O if and only if the residue class map HomRc(U, V)~ Homk6 (U, V) is epimorphic. P r o o f. Application of the long exact cohomology sequence yields the exact sequence 0 ~HomRc (U, V) ~ HomRo (U, V) --*Homko (0, ~2)-~Ext~o (U, V) Z*Ext~a (U, V) .... .

Finite soluble groups with supersoluble Sylow normalizers

Finite soluble groups with supersoluble Sylow normalizers

On the admissibility of finite groups over global fields of finite characteristic

Let k be a global field of characteristic p. A finite group G is called k-admissible if there exists a division algebra finite dimensional and central over k which is a crossed product for G. Let G be a finite group with normal Sylow p-subgroup P. If the factor group G P is k-admissible, then G is k-admissible. A necessary condition is given for a group to be k-admissible: if a finite group G is k-admissible, then every Sylow l-subgroup of G for l ≠ p is metacyclic with some additional restriction. Then it is proved that a metacyclic group G generated by x and y is k-admissible if some relation between x and y is satisfied.

The modular group algebra of certain finite groups with normal sylow p-subgroups

The modular group algebras we consider are of the form kP >a Q where P is a finite p-group, Q a finite p’ group, k a splitting field of characteristic p for Q, and the action of Q on P arises from a homomorphism a: Q + Aut P with abelian image A. (By the Schur-Zassenhaus theorem (cf. [3]), every finite group G with a normal Sylow p-subgroup P a G is a semidirect product P ~1 G/P; but of course the action need not be abelian.) Theorem 1.1 is a special case of Theorem 1.3, stated below, which gives sufficient conditions for an isomorphism of group algebras of the form kP >a Q. Theorem 1.3 in turn is an application of Theorem 1.2 which gives an explicit description of the blocks of such a group algebra. We note that we are dealing here with a class of blocks which do not in general have finite representation type. and indeed can have any p-group as defect group (the defect group is P). Theorem 1.2 can be regarded as lifting the Wedderburn theorem for kQ up to kP >a Q in accordance with the numerical data about blocks provided by Brauer’s theorem (cf. [ 1, p. 6341). It expresses kP ~1 Q as a direct sum of full

On finite groups with a certain Sylow normalizer

On finite groups with a certain Sylow normalizer

Simple groups with a Sylow normalizer of order 3 p

In this paper the following main theorem is proved. Theorem. Let G be a finite simple group with a Sylow p-normalizer in G of order 3p. If there is a nonprincipal ordinary irreducible character of degree at most 29 in the principal p-block, then G is isomorphic to one of the groups L(3, 2), A 7, U(3, 3), L(3, 3), C7(3, 4), L(3, 4), U(3, 5), or A 8.

5-fold transitive permutation groups in which the stabilizer of five points has a normal Sylow 2-subgroup

5-fold transitive permutation groups in which the stabilizer of five points has a normal Sylow 2-subgroup

A Note on Finite Groups Having a Fixed-Point-Free Automorphism

A fusion result of Glauberman has as a consequence the fact that a finite group admitting a fixed-point-free automorphism has a normal Sylow $2$-subgroup (and in particular, is solvable) if all nontrivial fixed-point subgroups have odd order.

On the structure of certain p-solvable linear groups

There are four sections in the paper. In the first section notation, past results and some preparations on p-solvable linear groups are presented. In the second section the inductive proof of the Theorem is begun and we reduce down to the case where G has a p-Sylow subgroup P of order p and a normal p-complement N, where N modulo the center is a nonabelian simple group. In Section 3, detailed information about some Sylow normalizers of N is obtained. In the last section the strong interplay between various Sylow normalizers of N and the subgroup %YN(P) is used in completing the proof. A recent result of Smith and Tyrer ([12]) helps dispose of some difficult cases that arise in Sections 3 and 4. The proof also apparently requires some deep classification theorems on finite simple groups in contrast to the earlier papers. Although we have not addressed ourselves to the problem of describing the structure of G when G is solvable, the following result can be extracted from the proof of the Theorem.

On finite groups with a certain Sylow normalizer. III

On finite groups with a certain Sylow normalizer. III

Finite Groups with Nicely Supplemented Sylow Normalizers

Finite Groups with Nicely Supplemented Sylow Normalizers

On finite groups with a certain sylow normalizer. I

On finite groups with a certain sylow normalizer. I