Elementary Abelian Subgroups Research Articles

On p-groups with a maximal elementary abelian normal subgroup of rank k

There are several results in the literature concerning p-groups G with a maximal elementary abelian normal subgroup of rank k due to Thompson, Mann and others. Following an idea of Sambale we obtain bounds for the number of generators etc. of a 2-group G in terms of k, which were previously known only for p>2. We also prove a theorem that is new even for odd primes. Namely, we show that if G has a maximal elementary abelian normal subgroup of rank k, then for any abelian subgroup A the Frattini subgroup Φ(A) can be generated by 2k elements (3k when p=2). The proof of this rests upon the following result of independent interest: If V is an n-dimensional vector space, then any commutative subalgebra of End(V) contains a zero algebra of codimension at most n.

The Margolis homology of the cohomology restriction from an extra-special group to its maximal elementary abelian subgroups

The Margolis homology of the cohomology restriction from an extra-special group to its maximal elementary abelian subgroups

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.

Elementary Abelian subgroups in 2-groups whose derived subgroups are Klein 4-groups

Let P be a finite 2-group. Let be a poset consisting of elementary abelian subgroups of rank at least 2. If the derived subgroup , we proved the spheres occurring in all have the same dimension.

2-элементы в группе автотопизмов полуполевой проективной плоскости

We investigate the well-known hypothesis of D.R. Hughes that the full collineation group of non-Desarguesian semifield projective plane of a finite order is solvable (the question 11.76 in Kourovka notebook was written down by N.D. Podufalov). The spread set method is used to construct the semifield projective planes with cyclic 2- subgroup of autotopisms in the case of linear space of any dimension over the field of prime order. This study completes the analogous considerations of elementary abelian 2-subgroups. We obtain the natural restriction to the order of 2-element for the semifield planes for odd and even order. It is proved that some projective linear groups can not be the autotopism subgroups for the infinite series of semifield planes. The matrix representation of Baer involution allows us to define the geometric property of autotopism of order 4. We can choose the base of a linear space such that the matrix representation of these autotopisms is convenient and uniform, it does not depend on the space dimension. The minimal counter-example is constructed to explain the restriction to the plane order. As a corollary, we proved the solvability of the full collineation group when the nonDesarguesian semifield plane has a certain even order and all its Baer subplanes are also non-Desarguesain. The main results can be used as technical for the further studies of the subgroups of even order in an autotopism group for a finite non-Desarguesian semifield plane. The results obtained are useful to investigate the semifield planes with the autotopism subgroups from J.G. Thompson’s list of minimal simple groups.

Commuting involutions and elementary abelian subgroups of simple groups

Motivated in part by representation theoretic questions, we prove that if G is a finite quasi-simple group, then there exists an elementary abelian subgroup of G that contains a member of each conjugacy class of involutions of G.

Influence of complemented subgroups on the structure of finite groups

P‎. ‎Hall proved that a finite group $G$ is supersoluble with elementary abelian Sylow subgroups if and only if every subgroup of $G$ is complemented in $G$‎. ‎He called such groups complemented‎. ‎A‎. ‎Ballester-Bolinches and X‎. ‎Guo‎ ‎established the structure of minimal non-complemented groups‎. ‎We give the classification of finite non-soluble groups‎ ‎all of whose second maximal subgroups are complemented groups‎. ‎We also prove that every finite group with less than‎ ‎$21$ non-complemented non-minimal ${2,3,5}$-subgroups is soluble‎.

Generalized Gardiner–Praeger graphs and their symmetries

A subgroup of the automorphism group of a graph acts half-arc-transitively on the graph if it acts transitively on the vertex-set and on the edge-set of the graph but not on the arc-set of the graph. If the full automorphism group of the graph acts half-arc-transitively, the graph is said to be half-arc-transitive.In 1994 Gardiner and Praeger introduced two families of tetravalent arc-transitive graphs, called the C±1 and the C±ε graphs, that play a prominent role in the characterization of the tetravalent graphs admitting an arc-transitive group of automorphisms with a normal elementary abelian subgroup such that the corresponding quotient graph is a cycle. All of the Gardiner–Praeger graphs are arc-transitive but admit a half-arc-transitive group of automorphisms. Quite recently, Potočnik and Wilson introduced the family of CPM graphs, which are generalizations of the Gardiner–Praeger graphs. Most of these graphs are arc-transitive, but some of them are half-arc-transitive. In fact, at least up to order 1000, each tetravalent half-arc-transitive loosely-attached graph of odd radius having vertex-stabilizers of order greater than 2 is isomorphic to a CPM graph.In this paper we determine the automorphism group of the CPM graphs and investigate isomorphisms between them. Moreover, we determine which of these graphs are 2-arc-transitive, which are arc-transitive but not 2-arc-transitive, and which are half-arc-transitive.

On the decomposability of mod 2 cohomological invariants of Weyl groups

We compute the invariants of Weyl groups in mod 2 Milnor K -theory and more general cycle modules, which are annihilated by 2. Over a base field of characteristic coprime to the group order, the invariants decompose as direct sums of the coefficient module. All basis elements are induced either by Stiefel–Whitney classes or specific invariants in the Witt ring. The proof is based on Serre’s splitting principle that guarantees detection of invariants on elementary abelian 2-subgroups generated by reflections.

On properties of translation groups in the affine general linear group with applications to cryptography

The affine general linear group acting on a vector space over a prime field is a well-understood mathematical object. Its elementary abelian regular subgroups have recently drawn attention in applied mathematics thanks to their use in cryptography as a way to hide or detect weaknesses inside block ciphers. This paper is focused on building a convenient representation of their elements which suits better the purposes of the cryptanalyst. Several combinatorial counting formulas and a classification of their conjugacy classes are given as well.

Groups with irreducibly unfaithful subsets for unitary representations

Let $G$ be a group. A subset $F \subset G$ is called irreducibly faithful if there exists an irreducible unitary representation $\pi$ of $G$ such that $\pi(x) \neq \mathrm{id}$ for all $x \in F \smallsetminus \{e\}$. Otherwise $F$ is called irreducibly unfaithful. Given a positive integer $n$, we say that $G$ has Property $P(n)$ if every subset of size $n$ is irreducibly faithful. Every group has $P(1)$, by a classical result of Gelfand and Raikov. Walter proved that every group has $P(2)$. It is easy to see that some groups do not have $P(3)$. We provide a complete description of the irreducibly unfaithful subsets of size $n$ in a countable group $G$ (finite or infinite) with Property $P(n-1)$: it turns out that such a subset is contained in a finite elementary abelian normal subgroup of $G$ of a particular kind. We deduce a characterization of Property $P(n)$ purely in terms of the group structure. It follows that, if a countable group $G$ has $P(n-1)$ and does not have $P(n)$, then $n$ is the cardinality of a projective space over a finite field. A group $G$ has Property $Q(n)$ if, for every subset $F \subset G$ of size at most $n$, there exists an irreducible unitary representation $\pi$ of $G$ such that $\pi(x) \ne \pi(y)$ for any distinct $x, y$ in $F$. Every group has $Q(2)$. For countable groups, it is shown that Property $Q(3)$ is equivalent to $P(3)$, Property $Q(4)$ to $P(6)$, and Property $Q(5)$ to $P(9)$. For $m, n \ge 4$, the relation between Properties $P(m)$ and $Q(n)$ is closely related to a well-documented open problem in additive combinatorics.

Mod-two cohomology rings of alternating groups

Abstract We calculate the direct sum of the mod-two cohomology of all alternating groups, with both cup and transfer product structures, which in particular determines the additive structure and ring structure of the cohomology of individual groups. We show that there are no nilpotent elements in the cohomology rings of individual alternating groups. We calculate the action of the Steenrod algebra and discuss individual component rings. A range of techniques are developed, including an almost Hopf ring structure associated to the embeddings of products of alternating groups and Fox–Neuwirth resolutions, which are new techniques. We also extend understanding of the Gysin sequence relating the cohomology of alternating groups to that of symmetric groups and calculation of restriction to elementary abelian subgroups.

On self-similar finite $p$-groups

In this paper, we address the following question: when is a finite p -group G self-similar, i.e. when can G be faithfully represented as a self-similar group of automorphisms of the p -adic tree? We show that, if G is a self-similar finite p -group of rank r , then its order is bounded by a function of p and r . This applies in particular to finite p -groups of a given coclass. In the particular case of groups of maximal class, that is, of coclass 1, we can fully answer the question above: a p -group of maximal class G is self-similar if and only if it contains an elementary abelian maximal subgroup over which G splits. Furthermore, in that case the order of G is at most p^{p+1} , and this bound is sharp.

Элементарные абелевы 2-подгруппы в группе автотопизмов полуполевой проективной плоскости

Элементарные абелевы 2-подгруппы в группе автотопизмов полуполевой проективной плоскости

Elementary abelian subgroups in some special 𝑝-groups

Abstract Let P be a finite p-group and p an odd prime. Let 𝒜 p ⁢ ( P ) ≥ 2 {\mathcal{A}_{p}(P)_{\geq 2}} be a poset consisting of elementary abelian subgroups of rank at least 2. If the derived subgroup P ′ ≅ C p × C p {P^{\prime}\cong C_{p}\times C_{p}} , then the spheres occurring in 𝒜 p ⁢ ( P ) ≥ 2 {\mathcal{A}_{p}(P)_{\geq 2}} all have the same dimension.

Regular subgroups with large intersection

In this paper we study the relationships between the elementary abelian regular subgroups and the Sylow $2$-subgroups of their normalisers in the symmetric group $\mathrm{Sym}(\mathbb{F}_2^n)$, in view of the interest that they have recently raised for their applications in symmetric cryptography.

A family of permutation groups with exponentially many nonconjugated regular elementary abelian subgroups

A family of permutation groups with exponentially many nonconjugated regular elementary abelian subgroups

Control of fusion by Abelian subgroups of the hyperfocal subgroup

We prove that an isomorphism between saturated fusion systems over the same finite [Formula: see text]-group is detected on the elementary abelian subgroups of the hyperfocal subgroup if [Formula: see text] is odd, and on the abelian subgroups of the hyperfocal subgroup of exponent at most [Formula: see text] if [Formula: see text]. For odd [Formula: see text], this has implications for mod [Formula: see text] group cohomology.

Reduced fusion systems over p-groups with abelian subgroup of index p: II

Let p be an odd prime, and let S be a p-group with a unique elementary abelian subgroup A of index p. We classify the simple fusion systems over all such groups S in which A is essential. The resulting list, which depends on the classification of finite simple groups, includes a large variety of new, exotic simple fusion systems.

The non-solvable doubly transitive dimensional dual hyperovals

AbstractIn [9] S. Yoshiara determines possible automorphism group of doubly transitive dimensional dual hyperovals. He shows that a doubly transitive dual hyperovalDis either isomorphic to the Mathieu dual hyperoval or the dual hyperoval is defined over 𝔽2, and if the hyperoval has rankn, the automorphism group has the formE⋅S, with an elementary abelian groupEof order 2nandSa subgroup of GL(n,2) acting transitively on the nontrivial elements ofE. Moreover Yoshiara describes the possible candidates forS. In this paper we assume thatSis non-solvable and show that then the dimensional dual hyperoval is a bilinear quotient of a Hyubrechts dual hyperoval.