P-subgroup Research Articles

On finite groups with certain maximal CSS-subgroups

Let G be a finite group. We say that a subgroup H of G is a CSS-subgroup in G, if there exists a normal subgroup K of G such that G = HK and H ∩ K is SS-quasinormal in G. In this paper, we study the p-nilpotency and supersolvability of G under the assumption that every member of a small subset of the set of all maximal subgroups of the Sylow p-subgroups P of G with certain property is a CSS-subgroup in G. Using this criterion, we obtain new results which generalize and unify recent results in the literature.

A new criterion for p-nilpotency and solvability of finite groups by coprime actions

Let G and A be finite groups of relative coprime orders and A acts on G via automorphisms. In this paper, we investigate the p-nilpotency and solvability of finite groups in which every maximal A-invariant subgroup of a Sylow p-subgroup of G for some prime p is S-semipermutable.

Modules with finitely generated cohomology

Let G be a finite group and k a field of characteristic p. It is conjectured in a paper of the first author and John Greenlees that the thick subcategory of the stable module category StMod(kG) consisting of modules whose cohomology is finitely generated over H⁎(G,k) is generated by finite dimensional modules and modules with no cohomology. If the centraliser of every element of order p in G is p-nilpotent, this statement follows from previous work. Our purpose here is to prove this conjecture in two cases with non p-nilpotent centralisers. The groups involved are Z/3r×Σ3 (r⩾1) in characteristic three and Z/2×A4 in characteristic two.As a consequence, in these cases the bounded derived category of C⁎BG (cochains on BG with coefficients in k) is generated by C⁎BS, where S is a Sylow p-subgroup of G.

On Oliver's p-group conjecture for Sylow subgroups of symplectic groups and orthogonal groups

In this paper, we focus on Oliver's p-group conjecture. We prove that Oliver's p-group conjecture holds for Sylow p-subgroups of symplectic groups and orthogonal groups.

Intrinsic Geometries of the Simple Group of Order 168

It is well known that, up to isomorphism, there is a unique simple group of order 168. The projective special linear groups PSL(2, 7) and PSL(3, 2) are concrete examples of such groups, and are therefore isomorphic. This isomorphism is not obvious, but it has been clarified from several viewpoints. Here we use purely group-theoretic methods—the Sylow theorems, Poincaré’s theorem, and the orbit-stabilizer theorem—to show that any simple group of order 168 has a natural action on the projective line over F 7 , so it is isomorphic to PSL(2, 7). We discuss an analogous action on the projective plane over F 2 described by Smith and Tabachnikova. Both geometries are constructed using conjugacy classes of maximal subgroups. This gives another proof of the uniqueness theorem and may dispel some of the air of mystery that often surrounds the isomorphism between PSL(2, 7) and PSL(3, 2).

Primes and conductors of characters

If G is a finite p-solvable group, P is a Sylow p-subgroup of G and χ∈Irr(G) is an irreducible complex character of degree not divisible by p, we prove that the field of values Q(χP) of the restriction of χ to P has index not divisible by p in the cyclotomic extension Q(e2πipf), where pf is the p-part of the conductor of χ. This proves the p-solvable case of the main conjecture in [4].

Skew-morphisms of elementary abelian 𝑝-groups

Abstract A skew-morphism of a finite group 𝐺 is a permutation 𝜎 on 𝐺 fixing the identity element, and for which there exists an integer-valued function 𝜋 on 𝐺 such that σ ⁢ ( x ⁢ y ) = σ ⁢ ( x ) ⁢ σ π ⁢ ( x ) ⁢ ( y ) \sigma(xy)=\sigma(x)\sigma^{\pi(x)}(y) for all x , y ∈ G x,y\in G . It is known that, for a given skew-morphism 𝜎 of 𝐺, the product of the left regular representation of 𝐺 with ⟨ σ ⟩ \langle\sigma\rangle forms a permutation group on 𝐺, called a skew-product group of 𝐺. In this paper, we study the skew-product groups 𝑋 of elementary abelian 𝑝-groups Z p n \mathbb{Z}_{p}^{n} . We prove that 𝑋 has a normal Sylow 𝑝-subgroup and determine the structure of 𝑋. In particular, we prove that Z p n ⊲ X \mathbb{Z}_{p}^{n}\lhd X if p = 2 p=2 and either Z p n ⊲ X \mathbb{Z}_{p}^{n}\lhd X or ( Z p n ) X ≅ Z p n − 1 (\mathbb{Z}_{p}^{n})_{X}\cong\mathbb{Z}_{p}^{n-1} if 𝑝 is an odd prime. As an application, for n ≤ 3 n\leq 3 , we prove that 𝑋 is isomorphic to a subgroup of the affine group AGL ⁢ ( n , p ) \mathrm{AGL}(n,p) and enumerate the number of skew-morphisms of Z p n \mathbb{Z}_{p}^{n} .

Finite Groups with Certain Weakly S-permutable Subgroups

Let G be a finite group. A subgroup H of G is said to be weakly S-permutable in G if G has a subnormal subgroup T such that G = HT and T ∩ H ≤ HsG, where HsG is the subgroup of H generated by all those subgroups of H which are S-permutable in G. In this paper, we prove the following: For a Sylow p-subgroup P of G (p > 2), suppose that P has a subgroup D such that 1 < |D| < |P| holds and all subgroups H of P with |H| = |D| are weakly S-permutable in G. Then, the commutator subgroup G ′ is p-nilpotent. We certainly belive that this result will improve and extend a current and classical theories in the literature.

On redundant Sylow subgroups

A Sylow p-subgroup P of a finite group G is called redundant if every p-element of G lies in a Sylow subgroup different from P. Generalizing a recent theorem of Maróti–Martínez–Moretó, we show that for every non-cyclic p-group P there exists a solvable group G such that P is redundant in G. Moreover, we answer several open questions raised by Maróti–Martínez–Moretó.

A criterion for a normal subgroup to be hypercentral based on class sizes

AbstractLet G be a finite group and N a normal subgroup of G. We prove that the knowledge of the sizes of the conjugacy classes of G that are contained in N and of their multiplicities provides information of N in relation to the structure of G. Among other results, we obtain a criterion to determine whether a Sylow p-subgroup of N lies in the hypercentre of G for a fixed prime p, and therefore, whether the whole subgroup N is hypercentral in G.

A note on blocks of finite groups with TI Sylow p-subgroups

A note on blocks of finite groups with TI Sylow p-subgroups

On locally compact groups of small topological entropy

We discuss the finiteness of the topological entropy of continuous endomorphims for some classes of locally compact groups. Firstly, we focus on the abelian case, imposing the condition of being compactly generated, and note an interesting behaviour of slender groups. Secondly, we remove the condition of being abelian and consider nilpotent periodic locally compact p-groups (p prime), reducing the computations to the case of Sylow p-subgroups. Finally, we investigate locally compact Heisenberg p-groups Hn(Qp)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb{H}_n (\\mathbb{Q}_ p )$$\\end{document} on the field Qp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb{Q}_ p $$\\end{document} of the p-adic rationals with n arbitrary positive integer.

Exploration of Symmetric Groups: Cayley Tables, Subgroup Analysis, and Real-World Applications in Card Tricks

This research delves into the extensive analysis of symmetric groups of various orders and degrees. We explore different representations of permutations, construct a Cayley table for the symmetric group of degree four, and systematically identify all its subgroups using Lagrange's theorem. An intriguing discovery unfolds as we demonstrate that the converse of the theorem countered by Sylow's theorem does not hold universally. Furthermore, we apply the concepts of permutations and product of disjoint cycles to address a real-world problem – the Card Trick game. This study amalgamates theoretical exploration with practical applications, showcasing the versatility and depth of symmetric group theory.

Control of transfer for odd primes

Suppose p is a prime and T is a non-identity finite p-group. We show that if p is odd, then there exist two non-identity characteristic subgroups K1(T) and K2(T) of T such that K1 and K2 jointly control transfer in every finite group G containing T as a Sylow p-subgroup, in the sense that the normalizers of K1(T) and K2(T) in G determine the largest abelian factor group of G that is a p-group. We also give a new example of a characteristic subgroup K(T) of T such that K controls transfer in G by itself if p⩾5.

Applications of conjunctive complex fuzzy subgroups to Sylow theory

<abstract><p>Sylow's theorems are fundamental theorems in classical group theory that are of paramount importance. The extension of these theorems into diverse fuzzy contexts emerges as a compelling area of exploration. This study introduces the novel concept of the conjunctive complex fuzzy conjugate element within the conjunctive complex fuzzy subgroup of a group, elucidating numerous crucial properties of this concept. Additionally, it propounds the notion of the conjunctive complex fuzzy <italic>p</italic>-subgroup within the conjunctive complex fuzzy subgroup (CCFSG) and delineates various indispensable characteristics associated with this construct. Additionally, the paper formulates the conjunctive complex fuzzy version of the Cauchy theorem for finite groups. Lastly, it defines the concept of the conjunctive complex fuzzy Sylow <italic>p</italic>-subgroup for a finite group and conducts a generalization of Sylow's theorems within a conjunctive complex fuzzy environment.</p></abstract>

Finite groups with many p-regular conjugacy classes

Let G be a finite group and let p be a prime. In this paper, we study the structure of finite groups with a large number of p-regular conjugacy classes or, equivalently, a large number of irreducible p-modular representations. We prove sharp lower bounds for this number in terms of p and the p′-part of the order of G which ensure that G is p-solvable. A bound for the p-length is obtained which is sharp for odd primes p. We also prove a new best possible criterion for the existence of a normal Sylow p-subgroup in terms of these quantities.

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.

Group action and its application on classification of low order groups

The classification of finite groups is an important topic in mathematics throughout history of mathematics. The topic of this paper is to use group action as a tool, to classify some special finite groups and some low order groups. First this paper introduces some concepts of group action. Then this paper states and proves some important theorems related to group action. For example, the Sylows theorem, which is very important in this paper. Research has found that, groups of specific order, such as groups whose order are 2p,p^2, pq(p, q are distinct prime numbers), p3(p is prime) can be classified using group action and the technique of semi-direct product, and groups whose order are no more than 15 are classified which can be seen as the special situations of the above ones. But in general, to make classification of a larger range of finite groups, more tools should be introduced.

Covering the set of p-elements in finite groups by Sylow p-subgroups

Let Gp be the set of p-elements of a finite group G. Do we need all the Sylow p-subgroups of G to cover Gp? Although this question does not have an affirmative answer in general, our work indicates that the answer is yes more often than one could perhaps expect.

A modular idealizer chain and unrefinability of partitions with repeated parts

Recently Aragona et al. have introduced a chain of normalizers in a Sylow 2-subgroup of Sym(2n), starting from an elementary abelian regular subgroup. They have shown that the indices of consecutive groups in the chain depend on the number of partitions into distinct parts and have given a description, by means of rigid commutators, of the first n − 2 terms in the chain. Moreover, they proved that the (n − 1)-th term of the chain is described by means of rigid commutators corresponding to unrefinable partitions into distinct parts. Although the mentioned chain can be defined in a Sylow p-subgroup of Sym(pn), for p > 2 computing the chain of normalizers becomes a challenging task, in the absence of a suitable notion of rigid commutators. This problem is addressed here from an alternative point of view. We propose a more general framework for the normalizer chain, defining a chain of idealizers in a Lie ring over ℤm whose elements are represented by integer partitions. We show how the corresponding idealizers are generated by subsets of partitions into at most m − 1 parts and we conjecture that the idealizer chain grows as the normalizer chain in the symmetric group. As evidence of this, we establish a correspondence between the two constructions in the case m = 2.