On 2-maximal subgroups of finite groups

Generation and random generation: From simple groups to maximal subgroups

On terraces for abelian groups

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

Properties of an arbitrary finite π-solvable group whose maximal subgroups are Hall subgroups are established.

On the maximal subgroups of finite simple groups

Finite simple nonabelian groups G that are not π-closed for some set of primes π but have π-closed maximal subgroups (property (*) for (G, π)) are studied. We give a list L of finite simple groups that contains any group G with the above property (for some π). It is proved that 2 ∉ π for any pair (G, π) with property (*) (Theorem 1). In addition, we specify for any sporadic simple group G from L all sets of primes π such that the pair (G, π) has property (*) (Theorem 2). The proof uses the author’s results on the control of prime spectra of finite simple groups.

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.

In [Li and Chen, A new characterization of the simple group [Formula: see text], Sib. Math. J. 53(2) (2012) 213–247.], it is proved that the simple group [Formula: see text] is uniquely determined by the set of orders of its maximal abelian subgroups. Also in [Momen and Khosravi, Groups with the same orders of maximal abelian subgroups as [Formula: see text], Monatsh. Math. 174 (2013) 285–303], the authors proved that if [Formula: see text], where [Formula: see text] is not a Mersenne prime, then every finite group with the same orders of maximal abelian subgroups as [Formula: see text], is isomorphic to [Formula: see text] or an extension of [Formula: see text] by a subgroup of the outer automorphism group of [Formula: see text]. In this paper, we prove that if [Formula: see text] is a finite group with the same orders of maximal abelian subgroups as [Formula: see text], then [Formula: see text] has a unique nonabelian composition factor which is isomorphic to [Formula: see text].

We prove in this paper that a finite almost simple group $R$ with socle the non-abelian simple group $S$ possesses a conjugacy class of maximal subgroups whose index coincides with the smallest index $\operatorname{l}(S)$ of a maximal group of $S$ or there exists a natural number $v_S \leq {\operatorname{l}(S)^2}$, depending only on $S$, such that $R$ has a conjugacy class of core-free maximal subgroups with index $v_S$. We show that the number of subgroups of the outer automorphism group of $S$ is bounded by $\log^3 {\operatorname{l}(S)}$ and $\operatorname{l}(S)^2 < |S|$.

Let G be a finite group with an abelian normal subgroup N. When does N have a unique conjugacy class of complements in G? We consider this question with a focus on properties of maximal subgroups. As corollaries we obtain Theorems 1.6 and 1.7 which are closely related to a result by Parker and Rowley on supplements of a nilpotent normal subgroup [3, Theorem 1]. Furthermore, we consider families of maximal subgroups of G closed under conjugation whose intersection equals Φ ⁢ ( G ) {\Phi(G)} . In particular, we characterize the soluble groups having a unique minimal family with this property (Theorem 2.3, Remark 2.4). In the case when Φ ⁢ ( G ) = 1 {\Phi(G)=1} , these are exactly the soluble groups in which each abelian normal subgroup has a unique conjugacy class of complements.

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.

Let p be a prime and the variety of elementary abelian by elementary abelian p-groups. A result of Brisley and Macdonald is generalized as follows. If H is a finite group in and G is a soluble group of p–power exponent such that no section of G is isomorphic to H, then G is nilpotent and its class is bounded by a function of three variables: H, the exponent of G, and the soluble length of G. It is a corollary that if the variety generated by a soluble group G of finite exponent contains , then each finite group in is isomorphic to some section of G.

In this paper, we study the number of conjugacy classes of maximal cyclic subgroups of a finite group 𝐺, denoted η ⁢ ( G ) \eta(G) . First we consider the properties of this invariant in relation to direct and semi-direct products, and we characterize the normal subgroups 𝑁 with η ⁢ ( G / N ) = η ⁢ ( G ) \eta(G/N)=\eta(G) . In addition, by applying the classification of finite groups whose nontrivial elements have prime order, we determine the structure of G / ⟨ G − ⟩ G/\langle G^{-}\rangle , where G − G^{-} is the set of elements of 𝐺 generating non-maximal cyclic subgroups of 𝐺. More precisely, we show that G / ⟨ G − ⟩ G/\langle G^{-}\rangle is either trivial, elementary abelian, a Frobenius group or isomorphic to A 5 A_{5} .

By a classical theorem of Jordan, every faithful transitive action of a non‐trivial finite group has a derangement (an element with no fixed points). The existence of derangements with additional properties has attracted much attention, especially for faithful primitive actions of almost simple groups. In this paper, we show that an almost simple group can have an element that is a derangement in every faithful primitive action, and we call these elements totally deranged. In fact, we classify the totally deranged elements of all almost simple groups, showing that an almost simple group contains a totally deranged element only if the socle of is or with . Using this, we classify the invariable generating sets of a finite simple group of the form where and , answering a question of Garzoni. As a final application, we classify the elements of almost simple groups that are contained in a unique maximal subgroup in the case where is not core‐free, which complements the recent work of Guralnick and Tracey addressing the case where is core‐free.

Let G be a connected noncompact simple Lie group with finite center and let A be a maximal compact subgroup of G. Assume the space G/K is Hermitian symmetric. We associate to each irreducible representation τ of K a principal series representation W(τ) and a (7-equivariant Szegό-type integral operator Sτ such that Sτ maps the K-finite vectors in W(τ) onto an irreducible highest weight $module L(τ). Of primary concern here are those representations τ which are reduction points. For such τ, we construct certain systems 3lτ of ΰ-equivariant differential operators and then utilize 3fτ to establish the infinitesimal irreducibility of the image of Sτ. 1. Introduction. Let G be a connected noncompact simple Lie group with finite center and let K be a maximal compact subgroup of G. Assume the space G/K is Hermitian symmetric. The main purpose of this article is to realize each irreducible highest weight representation of G as the image of a G-equivariant quotient map defined on principal series representations. To make this more precise, recall that each irreducible highest weight representation πτ of G is parametrized by an irreducible unitary representation τ of K. Let C°°(G, τ) denote the space of τ-covariant C°°-functions on G. We associate to τ a principal series representation W(τ) and a Szego map Sτ: W(τ) -> C°°(G, τ) having the property that the ^-finite vectors in W(τ) are mapped onto an irreducible g-module equivalent to the derived action of πτ. In the case of discrete series and limits thereof, this type of result was proved by Knapp and Wallach [16] in the general setting where G is a semisimple equirank Lie group with finite center. The main result here is that the irreducibility of the image of *S τ persists for all highest weight representations. Moreover, for certain τ called reduction points, the irreducibility of Image (Sτ) is proved by showing this space is annihilated by a system Qίτ of Gequivariant differential operators. The system 3τ somewhat parallels the role of the Schmid operator in the Knapp and Wallach result. The realization of distinguished representations as irreducible images of quotient maps is a recurring theme in the literature which