Finite P-group Research Articles

Some new characteristic p-functors and Thompson’s critical subgroup

Let p be a prime and P be a finite p-group. We define a new characteristic subgroup K ( P ) of P and show that K ( P ) is a critical subgroup of P. Secondly, we define several new characteristic subgroups K * ( P ) , K i ( P ) for each positive integer i and show that they enjoy some interesting and new properties similar to the those of critical subgroups. Lastly, for a solvable group G, we define a new characteristic subgroup Y ( G ) , which satisfies the analogue of Thompson’s result for solvable groups.

Finite p-groups whose non-normal cyclic subgroups have cyclic quotient groups in their normalizers

Finite p-groups whose non-normal cyclic subgroups have cyclic quotient groups in their normalizers

A note on 𝑑-maximal 𝑝-groups

Abstract A finite 𝑝-group 𝐺 is said to be 𝑑-maximal if d ⁢ ( H ) < d ⁢ ( G ) d(H)<d(G) for every subgroup H < G H<G , where d ⁢ ( G ) d(G) denotes the minimal number of generators of 𝐺. A similar definition can be formulated when 𝐺 is acted on by some group 𝐴. In this paper, we extend earlier results of Kahn and Laffey to this more general setting and we answer a question of Berkovich on minimal non-metacyclic 𝑝-groups.

Uniqueness of factorization for fusion-invariant representations

Given a saturated fusion system over a finite p-group S, we provide criteria to determine when uniqueness of factorization into irreducible -invariant representations holds. We use them to prove uniqueness of factorization when the order of S is at most p 3. We also describe an example where the monoid of fusion-invariant representations is not even half-factorial. Finally, we find other examples of fusion systems where this monoid is not factorial using GAP.

On cubic bi-Cayley graphs of p-groups

A graph is called a Cayley graph (or bi-Cayley graph, respectively) of a group G if it has a group G of automorphisms acting semiregularly on the vertices with exactly one orbit (or two orbits, respectively). It is known every Cayley graph is vertex-transitive. In this paper, we first present a classification of connected cubic non-Cayley vertex-transitive bi-Cayley graphs of a finite p-group H, where p > 3 is a prime and the derived subgroup of H is either cyclic or isomorphic to Zp × Zp. This is then used to give a classification of connected cubic non-Cayley vertex-transitive graphs of order 2p4 for each prime p.

The Amit–Ashurst conjecture for finite metacyclic p-groups

The Amit conjecture about word maps on finite nilpotent groups has been shown to hold for certain classes of groups. The generalised Amit conjecture says that the probability of an element occurring in the image of a word map on a finite nilpotent group G is either 0, or at least 1/|G|. Noting the work of Ashurst, we name the generalised Amit conjecture the Amit–Ashurst conjecture and show that the Amit–Ashurst conjecture holds for finite p-groups with a cyclic maximal subgroup.

On weak commutativity in 𝑝-groups

Abstract The weak commutativity group χ ⁢ ( G ) \chi(G) is generated by two isomorphic groups 𝐺 and G φ G^{\varphi} subject to the relations [ g , g φ ] = 1 [g,g^{\varphi}]=1 for all g ∈ G g\in G . We present new bounds for the exponent of χ ⁢ ( G ) \chi(G) and its sections, when 𝐺 is a finite 𝑝-group.

Baumann-components of finite groups of characteristic p, the W(B)-theorem

This paper completes the investigation of finite ▪-groups of characteristic p in terms of their Baumann components we began in [5] and [6].In this paper we define for each finite p-group B a non-trivial characteristic subgroup ▪ and for each finite ▪-group G of characteristic p with ▪, subnormal subgroups of G called Baumann blocks of G.We prove that ▪, where ▪ is the normal subgroup generated by the Baumann blocks of G. Moreover, we give the exact structure of the Baumann blocks of G and show that any two distinct Baumann blocks centralize each other.

The {{,mathrm{mathcal {X}},}}-series of a p-group and Complements of Abelian Subgroups

Let G be a finite p-group. We denote by {{,mathrm{mathcal {X}},}}_i(G) the intersection of all subgroups of G having index p^i in G. In this paper, the newly introduced series {{{,mathrm{mathcal {X}},}}_i(G)}_i is investigated and a number of results concerning its behaviour are proved. As an application of these results, we show that if an abelian subgroup A of G intersects each one of the subgroups {{,mathrm{mathcal {X}},}}_i(G) at {{,mathrm{mathcal {X}},}}_i(A), then A has a complement in G. Conversely if an arbitrary subgroup H of G has a normal complement, then {{,mathrm{mathcal {X}},}}_i(H) = {{,mathrm{mathcal {X}},}}_i(G) cap H.

Finite p-Groups G with H′=G′ for Each A2-Subgroup H

A finite [Formula: see text]-group [Formula: see text] is called an [Formula: see text]-group if [Formula: see text]is the minimal non-negative integer such that all subgroups of index [Formula: see text] of [Formula: see text] are abelian. The finite [Formula: see text]-groups [Formula: see text] with [Formula: see text] for all [Formula: see text]-subgroups [Formula: see text] of [Formula: see text]are classified completely in this paper. As an application, a problem proposed by Berkovich is solved.

On the occurrence of elementary abelian p-groups as the Schur multiplier of non-abelian p-groups

We prove that every elementary abelian p-group, for odd primes p, occurs as the Schur multiplier of some non-abelian finite p-group.

NONREALIZABILITY OF CERTAIN REPRESENTATIONS IN FUSION SYSTEMS

Abstract For a finite abelian p-group A and a subgroup $\Gamma \le \operatorname {\mathrm {Aut}}(A)$ , we say that the pair $(\Gamma ,A)$ is fusion realizable if there is a saturated fusion system ${\mathcal {F}}$ over a finite p-group $S\ge A$ such that $C_S(A)=A$ , $\operatorname {\mathrm {Aut}}_{{\mathcal {F}}}(A)=\Gamma $ as subgroups of $\operatorname {\mathrm {Aut}}(A)$ , and . In this paper, we develop tools to show that certain representations are not fusion realizable in this sense. For example, we show, for $p=2$ or $3$ and $\Gamma $ one of the Mathieu groups, that the only ${\mathbb {F}}_p\Gamma $ -modules that are fusion realizable (up to extensions by trivial modules) are the Todd modules and in some cases their duals.

Finite p-groups in which the intersection of nonnormal subgroup and center has bounded order

Given a non-negative integer i, a finite p-group G is called an -group if for every nonnormal subgroup H of G. It is proved that the center of -group has order or its derived subgroup is of order . And the classification of MC(p)-groups is obtained.

On equality of certain subcentral automorphisms in finite p-groups

Let [Formula: see text] be a group and [Formula: see text] be the group of all automorphisms of [Formula: see text]. In this paper, we study the structure of certain subgroups of the automorphisms group [Formula: see text] and give necessary and sufficient conditions for equality between certain subcentral automorphisms. As consequence of these results, we state a number of interesting corollaries about central automorphisms, autocentral automorphisms and derival automorphisms for finite [Formula: see text]-groups.

Finite p-groups with Few Kernels of Nonlinear Irreducible Characters

Let G be a finite nonabelian group and Kern(G) the set of kernels of nonlinear irreducible characters of G. In this paper, finite p-groups G with ∣Kern(G)∣ ≤ 3 are determined.

ON COMMUTATORS IN FINITE P-GROUPS OF ALMOST MAXIMAL CLASS

Let Γ(G) denote the set of commutators of a group G, G′ = ⟨Γ(G)⟩ and c(G) (or cw(G)) the minimal number such that every element of G′ can be expressed as a product of at most c(G) commutators. Recently we proved that if G is a finite p-group of maximal cl

On Group Invariants Determined by Modular Group Algebras: Even Versus Odd Characteristic

Let p be a an odd prime and let G be a finite p-group with cyclic commutator subgroup G′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$G^{\\prime }$\\end{document}. We prove that the exponent and the abelianization of the centralizer of G′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$G^{\\prime }$\\end{document} in G are determined by the group algebra of G over any field of characteristic p. If, additionally, G is 2-generated then almost all the numerical invariants determining G up to isomorphism are determined by the same group algebras; as a consequence the isomorphism type of the centralizer of G′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$G^{\\prime }$\\end{document} is determined. These claims are known to be false for p = 2.

Computational Analysis of Quantitative Characteristics of Some Residual Properties of Solvable Baumslag–Solitar Groups

Let Gk be defined as Gk = \(\left\langle {a,b;{{a}^{{ - 1}}}ba = {{b}^{k}}} \right\rangle \), where k ≠ 0. It is known that if p is a prime number then Gk is residually a finite p-group iff p|k – 1. It is also known that if p and q are primes not dividing k – 1, p < q and π = {p, q} then Gk is residually a finite π-group iff (k, q) = 1, p|q – 1 and the order of k in the multiplicative group of the field ℤq is a p-number. This paper examines the question of the number of two-element sets of prime numbers that satisfy the conditions of the last criterion. More precisely, let fk(x) be the number of sets {p, q} such that p < q, p \(\nmid \) k – 1, q \(\nmid \) k – 1, (k, q) = 1, p|q – 1, the order of k modulo q is a p-number, and p and q are chosen among the first x primes. We state that, if 2 ≤ |k| ≤ 10 000 and 1 ≤ x ≤ 50 000, then for almost all considered k the function fk(x) can be approximated quite accurately by the function αkx0.85, where the coefficient αk is different for each k and {αk|2 ≤ |k| ≤ 10 000} ⊆ (0.28; 0.31]. In addition, the dependence of the value fk(50 000) on k is investigated and an effective algorithm for checking a two-element set of prime numbers for compliance with the conditions of the last criterion is proposed. The results may have applications in the theory of computational complexity and algebraic cryptography.

Finite p-groups of class two with a large multiple holomorph

Let G be any group. The quotient group T(G) of the multiple holomorph by the holomorph of G has been investigated for various families of groups G. In this paper, we shall take G to be a finite p-group of class two for any odd prime p, in which case T(G) may be studied using certain bilinear forms. For any n≥4, we exhibit examples of G of order pn+(n2) such that T(G) contains a subgroup isomorphic toGLn(Fp)×GL(n2)−n(Fp). For finite p-groups G, the prime factors of the order of T(G) which were known so far all came from p(p−1). Our examples show that the order of T(G) can have other prime factors as well. In fact, we can embed any finite group into T(G) for a suitable choice of G.

Diagonal p-permutation functors, semisimplicity, and functorial equivalence of blocks

Let k be an algebraically closed field of characteristic p>0, let R be a commutative ring, and let F be an algebraically closed field of characteristic 0. We consider the R-linear category FRppkΔ of diagonal p-permutation functors over R. We first show that the category FFppkΔ is semisimple, and we give a parametrization of its simple objects, together with a description of their evaluations.Next, to any pair (G,b) of a finite group G and a block idempotent b of kG, we associate a diagonal p-permutation functor RTG,bΔ in FRppkΔ. We find the decomposition of the functor FTG,bΔ as a direct sum of simple functors in FFppkΔ. This leads to a characterization of nilpotent blocks in terms of their associated functors in FFppkΔ.Finally, for such pairs (G,b) of a finite group and a block idempotent, we introduce the notion of functorial equivalence over R, which (in the case R=Z) is slightly weaker than p-permutation equivalence, and we prove a corresponding finiteness theorem: for a given finite p-group D, there is only a finite number of pairs (G,b), where G is a finite group and b a block idempotent of kG with defect isomorphic to D, up to functorial equivalence over F.