Nontrivial P-subgroups Research Articles

Modular Conjectures for Direct Product of Finite Groups

The representation theory of a finite group, G, is an important area of research currently. This paper studied the modular representation of finite groups, which are direct products. There are three approaches to studying this representation: the ring approach, the character approach, and the module approach. Moreover, we learned some of the important conjectures in this representation, which link a representation of a finite group and its local subgroups, which are normalizer non-trivial p-subgroups. These conjectures are the McKay conjecture, Alperin’s weight conjecture, and the ordinary weight conjecture. The main aim of the proposed paper was to investigate these conjectures of direct products, the direct summands of which satisfy these conjectures for the associated tensor product of the p-block. We obtained the results by assuming the conjectures are true. Then, we used the properties of the direct products.

The global structure theorem for finite groups with an abelian large p-subgroup

For a prime p, the Local Structure Theorem [15] studies finite groups G with the property that a Sylow p-subgroup S of G is contained in at least two maximal p-local subgroups. Under the additional assumptions that G contains a so called large p-subgroup Q≤S, and that composition factors of the normalizers of non-trivial p-subgroups are from the list of the known simple groups, [15] partially describes the p-local subgroups of G containing S, which are not contained in NG(Q). In the Global Structure Theorem, we extend the work of [15] and describe NG(Q) and, in almost all cases, the isomorphism type of the almost simple subgroup H generated by the p-local over-groups of S in G. Furthermore, for p=2, the isomorphism type of G is determined. In this paper, we provide a reduction framework for the proof of the Global Structure Theorem and also prove the Global Structure Theorem when Q is abelian.

Endotrivial representations of finite groups and equivariant line bundles on the Brown complex

We relate endotrivial representations of a finite group in characteristic p to equivariant line bundles on the simplicial complex of non-trivial p-subgroups, by means of weak homomorphisms.

A generalisation of a theorem of Wielandt

In 1974, Helmut Wielandt proved that in a finite group G, a subgroup A is subnormal if and only if it is subnormal in 〈A,g〉 for all g∈G. In this paper, we prove that the subnormality of an odd order nilpotent subgroup A of G is already guaranteed by a seemingly weaker condition: A is subnormal in G if for every conjugacy class C of G there exists c∈C for which A is subnormal in 〈A,c〉. We also prove the following property of finite non-abelian simple groups: if A is a subgroup of odd prime order p in a finite almost simple group G, then there exists a cyclic p′-subgroup of F⁎(G) which does not normalise any non-trivial p-subgroup of G that is generated by conjugates of A.

Some Results on p-Nilpotence and Supersolvability of Finite Groups

Let G be a finite group. A subgroup K of a group G is called an ℋ-subgroup of G if N G (K) ∩ K x ≦ K for all x ∈ G. The set of all ℋ-subgroups of G will be denoted by ℋ(G). Let P be a nontrivial p-group. A chain of subgroups 1 = P 0 ≨ P 1 ≨ ··· ≨ P n = P is called a maximal chain of P provided that |P i : P i−1| = p, i = 1, 2, ···, n. A nontrivial p-subgroup P of G is called weakly supersolvably embedded in G if P has a maximal chain 1 = P 0 ≨ P 1 ≨ ··· ≨ P i ≨ ··· ≨ P n = P such that P i ∈ ℋ(G) for i = 1, 2, ···, n. Using the concept of weakly supersolvably embedded, we obtain new characterizations of p-nilpotent and supersolvable finite groups.

ON A $p$-LOCAL GEOMETRY ASSOCIATED WITH A COMPLEX OF NON-TRIVIAL $p$-SUBGROUPS

Let D be a complex of non-trivial p-subgroups of a finite group G, closed under G-conjugation. In this paper, a p-local geometry δp(G; D) is introduced for G associated with the complex D. The homotopy equivalence between δp(G; D) and D is also studied. 2000 Mathematics Subject Classification 20E15, 51E24.

The Radical 2-Subgroups of the Sporadic Simple Groups J4, Co2, and Th

For a finite group G and a prime divisor p of its order, a nontrivial Ž Ž .. p-subgroup R of G is called radical, if O N R R. With the poset p G Ž . B G of radical p-subgroups of G with respect to inclusion we naturally p Ž Ž .. associate the simplicial complex of its chains, denoted B G , which is p known to be G-homotopy equivalent to the complex associated with the poset of nontrivial p-subgroups as well as that of nontrivial elementary Ž . abelian p-subgroups of G see, e.g., 5, 6.6 . The latter are important tools Ž to investigate the structure of G concerning the prime p see, e.g., 5, . Ž Ž .. Chap. 6 . Thus the smaller complex B G has enough information to p understand behaviors of G on p. The following fact, a corollary of the Borel Tits theorem 5, 6.8.4 , Ž Ž .. shows the further importance of B G : for a finite group of Lie type p Ž Ž .. defined over a field in characteristic p, the complex B G is the p barycentric subdivision of the building for G. Thus for an arbitrary finite Ž Ž .. group G we may think of the complex B G as a generalization of the p concept of buildings: in fact, as is observed in 14 , for each sporadic simple Ž Ž .. group G of characteristic-p type, B G is homotopy equivalent to p some smaller simplicial complex, previously constructed by an ad hoc method and referred to as a ‘‘p-local geometry’’ of G. This suggests the importance of examining minimal complexes which are homotopy equivaŽ Ž .. Ž . Ž lent to B G for every specifically sporadic simple group G see also p . the introduction of 17 .

Homotopy properties of the poset of nontrivial p-subgroups of a group

Homotopy properties of the poset of nontrivial p-subgroups of a group