Elementary Abelian P-group Research Articles

The homomorphisms between the Dickson–Mùi algebras as modules over the Steenrod algebra

The Dickson–Mui algebra consists of all invariants in the mod p cohomology of an elementary abelian p-group under the general linear group. It is a module over the Steenrod algebra, \({\mathcal {A}}\) . We determine explicitly all the \({\mathcal {A}}\) -module homomorphisms between the (reduced) Dickson–Mui algebras and all the \({\mathcal {A}}\) -module automorphisms of the (reduced) Dickson–Mui algebras. The algebra of all \({\mathcal {A}}\) -module endomorphisms of the (reduced) Dickson–Mui algebra is claimed to be isomorphic to a quotient of the polynomial algebra on one indeterminate. We prove that the reduced Dickson–Mui algebra is atomic in the meaning that if an \({\mathcal {A}}\) -module endomorphism of the algebra is non-zero on the least positive degree generator, then it is an automorphism. This particularly shows that the reduced Dickson–Mui algebra is an indecomposable \({\mathcal {A}}\) -module. The similar results also hold for the odd characteristic Dickson algebras. In particular, the odd characteristic reduced Dickson algebra is atomic and therefore indecomposable as a module over the Steenrod algebra.

Modules of Constant Jordan Type with Small Non-Projective Part

Let E be an elementary abelian p-group of rank r and let k be an algebraically closed field of characteristic p. We prove that if M is a kE-module of stable constant Jordan type [a 1]...[a t ] with ∑ j a j ≤ min(r − 1, p − 2) then a 1 = ... = a t = 1. The proof uses the theory of Chern classes of vector bundles on projective space.

Elementary abelian p-groups of rank 2p+3 are not CI-groups

For every prime p>2 we exhibit a Cayley graph on $\mathbb{Z}_{p}^{2p+3}$ which is not a CI-graph. This proves that an elementary abelian p-group of rank greater than or equal to 2p+3 is not a CI-group. The proof is elementary and uses only multivariate polynomials and basic tools of linear algebra. Moreover, we apply our technique to give a uniform explanation for the recent works of Muzychuk and Spiga concerning the problem.

FINITE SELF-SIMILAR p-GROUPS WITH ABELIAN FIRST LEVEL STABILIZERS

We determine all finite p-groups that admit a faithful, self-similar action on the p-ary rooted tree such that the first level stabilizer is abelian. A group is in this class if and only if it is a split extension of an elementary abelian p-group by a cyclic group of order p.The proof is based on use of virtual endomorphisms. In this context the result says that if G is a finite p-group with abelian subgroup H of index p, then there exists a virtual endomorphism of G with trivial core and domain H if and only if G is a split extension of H and H is an elementary abelian p-group.

Characterization of p-schemes of prime cube order

Let p be a prime, let S be a non-commutative p-scheme of order p 3 , and assume that S is not thin. We prove that each closed subset of order p 2 of S is an elementary abelian p-group. We also construct non-commutative schurian p-schemes of order p 3 .

Signed permutation modules, Singer cycles and class numbers

Let p be a rational prime. The k(GV) theorem states that, given a finite p′-group G acting faithfully on a finite elementary abelian p-group V, the number of conjugacy classes of the semidirect product GV is bounded above by the order of V (k(GV) ⩽ |V|). In the present paper we examine when the upper bound k(GV) = |V| is attained. It is shown that for p > 5 this happens if and only if G/CG(U) ≌ CG(V/U) is cyclic of order |U| – 1 for each nontrivial irreducible submodule U of V (Singer cycle). It remains open whether this is also true when p = 5. For p = 2, 3 there exist examples where equality holds but G is not abelian.

The homomorphisms between the Dickson–Mùi algebras as modules over the Steenrod algebra

The Dickson–Mùi algebra consists of all invariants in the mod p cohomology of an elementary abelian p-group under the general linear group. It is a module over the Steenrod algebra, A . We determine explicitly all the A -module homomorphisms between the Dickson–Mùi algebras and all the A -module automorphisms of these algebras.

On the Number of Generators of a Bieberbach Group

In this article, we study the question of whether it is true that an n-dimensional Bieberbach group can be generated by n elements. We answer this question affirmatively in dimension less than or equal to 6, and in case the holonomy group is an elementary abelian p-group. This result includes an alternative proof for the recent result by Yalçin, saying that whenever an elementary abelian p-group is acting freely on the n-dimensional torus, we must have that h ≤ n.

On the order of Schur multiplier of non-abelian p-groups

Let G be a finite p-group of order p n , Green proved that M ( G ) , its Schur multiplier is of order at most p 1 2 n ( n − 1 ) . Later Berkovich showed that the equality holds if and only if G is elementary abelian of order p n . In the present paper, we prove that if G is a non-abelian p-group of order p n with derived subgroup of order p k , then | M ( G ) | ⩽ p 1 2 ( n + k − 2 ) ( n − k − 1 ) + 1 . In particular, | M ( G ) | ⩽ p 1 2 ( n − 1 ) ( n − 2 ) + 1 , and the equality holds in this last bound if and only if G = H × Z , where H is extra special of order p 3 and exponent p, and Z is an elementary abelian p-group.

THE CENTRE OF THE MAXIMAL p-SUBGROUP OF (pkD2p)

AbstractThe centre of the maximal p-subgroup of (pkD2p) is described as an elementary abelian p-group, where p is a prime.

The socle of a projective Mackey functor for a p-group

We study the simple subfunctors of indecomposable projective Mackey functors for a p-group P. Unlike the case of group algebras, the Mackey algebra is not in general self-injective. Thus, the socle of an indecomposable projective functor is not in general simple. We first show that the simple subfunctors of a projective functor P H , k , where H ⩽ P , are indexed by the normalizer in H of a subgroup of H. We then study the socle of a specific projective Mackey functor, namely the Burnside functor B P , and we focus on the case where P is abelian. In particular, our study enables us to determine the socle of an indecomposable projective Mackey functor indexed by a cyclic p-group, an abelian p-group of rank 2 and an elementary abelian p-group of rank 3.

Complexity and cohomology of cohomological Mackey functors

Let k be a field of characteristic p > 0 . Call a finite group G a poco group over k if any finitely generated cohomological Mackey functor for G over k has polynomial growth. The main result of this paper is that G is a poco group over k if and only if the Sylow p-subgroups of G are cyclic, when p > 2 , or have sectional rank at most 2, when p = 2 . A major step in the proof is the case where G is an elementary abelian p-group. In particular, when p = 2 , all the extension groups between simple functors can be determined completely, using a presentation of the graded algebra of self extensions of the simple functor S 1 G , by explicit generators and relations.

A theorem about coprime action

It is well known that if an elementary abelian p-group P acts on a p ′ -group Q and Q = [ Q , P ] , then Q = 〈 [ C Q ( A ) , P ] | A ⩽ P of index p〉. Does a similar statement hold for C Q ( P ) ? Under further assumptions, the answer is yes. Goldschmidt proves theorems of this flavour in [D.M. Goldschmidt, Weakly embedded 2-local subgroups of finite groups, J. Algebra 21 (1972) 341–351. [1]; D.M. Goldschmidt, Strongly closed 2-subgroups of finite groups, Ann. of Math. (2) 102 (1975) 475–489. [2]] and uses them to construct signalizer functors. For the same reason we prove a result of this type, under the assumption that Q is soluble.

RESTRICTING GLOBALLY-DEFINED MACKEY FUNCTORS TO NILPOTENT FINITE GROUPS

Let B be the Burnside ring considered as a globally-defined Mackey functor, and k be a field of positive characteristic q. We prove that, when k ⊗ℤ B is restricted to nilpotent groups of order prime to q, the only simple Mackey functors that can appear as subquotients of it are of the form SE,k, where E is the direct product of elementary abelian p-groups. In the special case k = 𝔽2, the simple Mackey functor SE,𝔽2 must appear as a filtration factor in 𝔽2 ⊗ℤ B for every direct product of elementary abelian p-groups E of odd order.

On the number of fuzzy subgroups of finite abelian groups

The main goal of this paper is to give an explicit formula for the number of fuzzy subgroups of a finite cyclic group. By using a similar method, some important steps in counting the number of fuzzy subgroups of a finite elementary abelian p-group are also made.

Semisymmetric graphs

Let ℘ N : X ˜ → X be a regular covering projection of connected graphs with the group of covering transformations isomorphic to N. If N is an elementary abelian p-group, then the projection ℘ N is called p-elementary abelian. The projection ℘ N is vertex-transitive (edge-transitive) if some vertex-transitive (edge-transitive) subgroup of the automorphism group of X lifts along ℘ N , and semisymmetric if it is edge- but not vertex-transitive. The projection ℘ N is minimal semisymmetric if it cannot be written as a composition ℘ N = ℘ ℘ M of two (nontrivial) regular covering projections, where ℘ M is semisymmetric. Malnič et al. [Semisymmetric elementary abelian covers of the Möbius–Kantor graph, Discrete Math. 307 (2007) 2156–2175] determined all pairwise nonisomorphic minimal semisymmetric elementary abelian regular covering projections of the Möbius–Kantor graph, the Generalized Petersen graph GP ( 8 , 3 ) , by explicitly giving the corresponding voltage rules generating the covering projections. It was remarked at the end of the above paper that the covering graphs arising from these covering projections need not themselves be semisymmetric (a graph with regular valency is said to be semisymmetric if its automorphism group is edge- but not vertex-transitive). In this paper it is shown that all these covering graphs are indeed semisymmetric.

Some Hopf Galois structures arising from elementary abelian $p$-groups

Let p be an odd prime, G =Z m p , the elementary abelian p-group of rank m, and let Γ be the group of principal units of the ring Fp [x]/(x m+1 ). If L/K is a Galois extension with Galois group Γ, then we show that for p > 5, the number of Hopf Galois structures on L/K afforded by K-Hopf algebras with associated group G is greater than p s , where s = (m-1) 2 3- m.

Hopf orders via Breuil modules

Let R be a discrete valuation ring with quotient field K and residue field k of characteristic p > 2 . Each finite flat abelian R-Hopf algebra of rank p n has a corresponding Breuil module. We determine the Breuil modules for the Hopf algebras which are generically isomorphic to KG where G is an elementary abelian p-group, and give an explicit classification for p > 3 , n ⩽ 2 .

An answer to Hirasaka and Muzychuk: Every p-Schur ring over [formula omitted] is Schurian

In Hirasaka and Muzychuk [An elementary abelian group of rank 4 is a CI-group, J. Combin. Theory Ser. A 94 (2) (2001) 339–362] the authors, in their analysis on Schur rings, pointed out that it is not known whether there exists a non-Schurian p-Schur ring over an elementary abelian p-group of rank 3. In this paper we prove that every p-Schur ring over an elementary abelian p-group of rank 3 is in fact Schurian.

Elementary Abelian p-groups of rank greater than or equal to 4p−2 are not CI-groups

In this paper we prove that an elementary Abelian p-group of rank 4p−2 is not a CI(2)-group, i.e. there exists a 2-closed transitive permutation group containing two non-conjugate regular elementary Abelian p-subgroups of rank 4p−2, see Hirasaka and Muzychuk (J. Comb. Theory Ser. A 94(2), 339–362, 2001). It was shown in Hirasaka and Muzychuk (loc cit) and Muzychuk (Discrete Math. 264(1–3), 167–185, 2003) that this is related to the problem of determining whether an elementary Abelian p-group of rank n is a CI-group. As a strengthening of this result we prove that an elementary Abelian p-group E of rank greater or equal to 4p−2 is not a CI-group, i.e. there exist two isomorphic Cayley digraphs over E whose corresponding connection sets are not conjugate in Aut E.