Endotrivial modules over groups with quaternion or semi-dihedral Sylow 2-subgroup

Variants of some of the Brauer-Fowler theorems

1.1. Let G be a connected reductive algebraic group defined over a finite (prime) field Fp of p elements. For a given rational G-module M one can consider the restrictions of M to the restricted Lie algebra Lie(G) and to the finite group G(Fp) of Fp-rational points. Even though there is no direct functorial connection between the categories of restricted Lie(G)-modules and G(Fp)-modules, the ambient algebraic group G can serve as a bridge linking these two important module categories. Early results of Curtis in the 1960s provided a one-to-one correspondence between the simple restricted modules (over k = Fp) of Lie(G) and those of G(Fp). This correspondence is given by simply restricting certain classes of simple G-modules. More importantly, these results, along with work of Steinberg, allow one to easily transfer the questions pertaining to the computation of irreducible characters among these three categories. For over thirty years, this approach has been one of the main ideas used to relate the representation and cohomology theory of algebraic groups, Lie algebras, and finite groups of Lie type (in the defining characteristic) (see, e.g., [CPSvan], [Jan2], [Hum3], and [Jan1]). Given the previous results in this area, a natural course of investigation should be to find relationships between canonical modules, like projective modules, for Lie(G) and G(Fp). More precisely,

Let G be a finite group. If G has a cyclic Sylow 2-subgroup, then G has the same number of irreducible rational-valued characters as of rational conjugacy classes. These numbers need not be the same even if G has Klein Sylow 2-subgroups and a normal 2-complement.

Let A be an abelian group. If for any we define , then is a quandle called the Takasaki quandle and is denoted by T(A). In this article, we focus on the lattice of subquandles of a finite Takasaki quandle T(A), denoted by . Indeed, we characterize the subquandles of T(A) and show that is the lattice of cosets of A if and only if A is an odd abelian group or its 2-Sylow subgroup is a cyclic group. Further, we determine the homotopy type of the lattice of subquandles of T(A) which is the wedge of r spheres of dimension d, where r is the number of prime numbers dividing and d is determined by applying the fundamental theorem of finite abelian groups. Furthermore, we prove that is graded if and only if A is an odd abelian group or its 2-Sylow subgroup is cyclic or its 2-Sylow subgroup is elementary abelian. Moreover, we show that the lattice of subquandles of a finite Takasaki quandle is determined by a unique Takasaki quandle. In other words, we show that the lattices of subquandles of two finite Takasaki quandles T(A) and T(B) are isomorphic if and only if A and B are isomorphic. However, the lattice of subquandles of a quandle which is not a Takasaki quandle can be isomorphic to the lattice of subquandles of a Takasaki quandle. We show this by providing a quandle Q of order 8 which is not a Takasaki quandle, but is isomorphic to

Overgroup coset complexes in finite groups

1 Introduction and Preliminaries All groups considered in this note will be finite. Recall that a minimal subgroup of a finite group is a subgroup of prime order. Many authors have investigated the structure of a finite group G, under the assumption that all minimal subgroups of G are well-situated in the group. Itô [7;III, 5.3] proved that if G is a group of odd order and all minimal subgroups of G lie in the center of G, then G is nilpotent. An extension of Itô's result is the following statement [7;IV,p.435]: If for an odd prime p, every subgroup of G of order p lies in the center of G, then G is p-nilpotent. If all element of G of orders 2 and 4 lie in the center of G, then G is 2-nilpotent. Buckley [4] proved that if G is a group of odd order and all minimal subgroups of G are normal in G, then G is supersoluble. Later Shaalan [8] proved that if G is a finite group and every subgroup of G of prime order or order 4 is π-quasinormal in G, then G is supersoluble. Recall that a subgroup H of a group G is π-quasinormal in G if H permutes with every Sylow subgroup of G. More recently, Shirong [9] proved that if the finite group G possesses a normal subgroup N of odd order such that G/N is supersoluble, and if, for each Sylow subgroup P of N, every minimal subgroup of P is normal in NG (P), then G is supersoluble. In [10] and [11] Yokoyama extends the results of Itô and Buckley for soluble groups using formation theory to generalize the notion of centrality.

Let a prime $p$ divide the order of a finite real reflection group. We classify the reflection subgroups up to conjugacy that are minimal with respect to inclusion, subject to containing a $p$-Sylow subgroup. For Weyl groups, this is achieved by an algorithm inspired by the Borel–de Siebenthal algorithm. The cases where there is not a unique conjugacy class of reflection subgroups minimally containing the $p$-Sylow subgroups are the groups of type $F_{4}$ when $p=2$ and $I_{2}(m)$ when $m\geq 6$ is even but not a power of $2$ for each odd prime divisor $p$ of $m$. The classification significantly reduces the cases required to describe the $p$-Sylow subgroups of finite real reflection groups.

Maximal 2-signalizers and centralizers of Sylow 2-subgroups in all finite simple groups are described. Also normalizers are computed for Sylow 2-subgroups in the finite simple groups of exceptional Lie type over a field of odd characteristic.

We have called a group admissible if it admits complete mappings and we proved that a finite group cannot be admissible if its Sylow 2-subgroup is nontrivial and cyclic. We proved the converse, known as the Hall-Paige conjecture, for abelian groups and groups of odd order. In this chapter we will prove the Hall-Paige conjecture true for more classes of groups. We will introduce HP-systems, an important tool in the construction of complete mappings, and use these to prove the Hall-Paige conjecture true for alternating and symmetric groups. We will also prove the Hall-Paige conjecture true for solvable groups, for Mathieu groups, for Suzuki groups, for certain unitary groups, and for groups whose Sylow 2-subgroups intersect trivially.

We study some families of finite groups having inner class-preservingautomorphisms. In particular, let G be a finite group and S be asemidihedral Sylow 2-subgroup. Then, in both cases when either Sym(4) is nota homomorphic image of G and Z(S)<Z(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Z(S) < Z(G)$$\\end{document} or G is nilpotent-by-nilpotent, wehave that all the Coleman automorphisms of G are inner. As a consequence, thesegroups satisfy the normalizer problem.

All groups considered are finite. A group has a trivial Frattini subgroup if and only if every nontrivial normal subgroup has a proper supplement.The property is normal subgroup closed, but neither subgroup nor quotient closed. It is subgroup closed if and only if the group is elementary, i.e. all Sylow subgroups are elementary abelian. If G is solvable, then G and all its quotients have trivial Frattini subgroup if and only if every normal subgroup of G has a complement. For a nilpotent group, every nontrivial normal subgroup has a supplement if and only if the group is elementary abelian. Consequently, the center of a group in which every normal subgroup has a supplement is an elementary abelian direct factor.

Let G be a finite group and H a subgroup of G. Then H is said to be S-quasinormal in G if HP = PH for all Sylow subgroups P of G. Let HsG be the subgroup of H generated by all those subgroups of H which are S-quasinormal in G. Then we say that H is nearly S-quasinormal in G if G has an S-quasinormal subgroup T such that HT = G and T ∩ H ≤ HsG. Our main result here is the following theorem. Let [Formula: see text] be a saturated formation containing all supersoluble groups and G a group with a normal subgroup E such that [Formula: see text]. Suppose that every non-cyclic Sylow subgroup P of E has a subgroup D such that 1 &lt; |D| &lt; |P| and all subgroups H of P with order |H| = |D| and every cyclic subgroup of P with order 4 (if |D| = 2 and P is a non-abelian 2-group) having no supersoluble supplement in G are nearly S-quasinormal in G. Then [Formula: see text].

Let G be a finite group and χ be an irreducible character of G. An efficient and simple method to construct representations of finite groups is applicable whenever G has a subgroup H such that χH has a linear constituent with multiplicity 1. In this paper we show (with a few exceptions) that if G is a simple group or a covering group of a simple group and χ is an irreducible character of G of degree less than 32, then there exists a subgroup H (often a Sylow subgroup) of G such that χH has a linear constituent with multiplicity 1.

The purpose of this paper is to study the subgroup embedding properties of S-semipermutability, semipermutability, and seminormality. Here we say H is S-semipermutable (resp. semipermutable) in a group Gif H permutes which each Sylow subgroup (resp. subgroup) of G whose order is relatively prime to that of H. We say H is seminormal in a group G if H is normalized by subgroups of G whose order is relatively prime to that of H. In particular, we establish that a seminormal p-subgroup is subnormal. We also establish that the solvable groups in which S-permutability is a transitive relation are precisely the groups in which the subnormal subgroups are all S-semipermutable. Local characterizations of this result are also established.

Given a finite group $G,$ let $e(G)$ be the expected number of elements of $G$ which have to be drawn at random, with replacement, before a set of generators is found. If all of the Sylow subgroups of $G$ can be generated by $d$ elements, then $e(G)\leq d+\kappa $, where $\kappa $ is an absolute constant that is explicitly described in terms of the Riemann zeta function and is the best possible in this context. Approximately, $\kappa $ equals 2.752394. If $G$ is a permutation group of degree $n,$ then either $G={Sym} (3)$ and $e(G)=2.9$ or $e(G)\leq \lfloor n/2\rfloor +\kappa ^*$ with $\kappa ^* \sim 1.606695.$ These results improve weaker bounds recently obtained by Lucchini.