Elementary Abelian P-group Research Articles

On monomial codes in modular group algebras

Let p be a prime number and K be the finite field of p elements, i.e. K=GF(p). Further let G be an elementary abelian p-group of order pm. Then the group algebra K[G] is modular. We consider K[G] as an ambient space and the ideals of K[G] as linear codes. A basis of a linear space is called visible, if there exists a member of the basis with the minimum (Hamming) weight of the space. The group algebra approach enables us to find some linear codes with a visible basis in the Jacobson radical of K[G]. These codes can be generated by “monomials” (Drensky & Lakatos, 1989). For p>2, some of our monomial codes have better parameters than the Generalized Reed–Muller codes. In the last part of the paper we determine the automorphism groups of some of the introduced codes.

Normal series and character values in p-standard table algebras

ABSTRACTWe investigate the character values and structures of p-standard table algebras (A,B) with o(B) = pN. If N≤3, then B has a complete normal series. If for every χ∈Irr(B), χ has at most p distinct classes of character values, and if either B has a complete normal series or p = 2, then B is an elementary abelian p-group.

Pseudo-free families of finite computational elementary abelian p-groups

AbstractWe initiate the study of (weakly) pseudo-free families of computational elementary abelian

Rings of invariants for the three-dimensional modular representations of elementary abelian p-groups of rank four

We show that the rings of invariants for the three dimensional modular representations of an elementary abelian $p$-group of rank four are complete intersections with embedding dimension at most five. Our results confirm the conjectures of Campbell, Shank and Wehlau.

Modules of constant Jordan type, pullbacks of bundles and generic kernel filtrations

Let kE denote the group algebra of an elementary abelian p-group of rank r over an algebraically closed field of characteristic p. We investigate the functors Fi from kE-modules of constant Jordan type to vector bundles on Pr−1(k), constructed by Benson and Pevtsova. For a kE-module M of constant Jordan type, we show that restricting the sheaf Fi(M) to a dimension s−1 linear subvariety of Pr−1(k) is equivalent to restricting M along a corresponding rank s shifted subgroup of kE and then applying Fi.In the case r=2, we examine the generic kernel filtration of M in order to show that Fi(M) may be computed on certain subquotients of M whose Loewy lengths are bounded in terms of i. More precise information is obtained by applying similar techniques to the nth power generic kernel filtration of M. The latter approach also allows us to generalise our results to higher ranks r.

Relation Between Groups with Basis Property and Groups with Exchange Property

Abstract A group G is called a group with basis property if there exists a basis (minimal generating set) for every subgroup H of G and every two bases are equivalent. A group G is called a group with exchange property, if x∉〈X〉 ⋀ x∈〈X∪{y}〉, then y∈〈X∪{x}〉, for all x, y ∈ G and for every subset X⊆G. In this research, we proved the following: Every polycyclic group satisfies the basis property. Every element in a group with the exchange property has a prime order. Every p-group satisfies the exchange property if and only if it is an elementary abelian p-group. Finally, we found necessary and sufficient condition for every group to satisfy the exchange property, based on a group with the basis property.

Minimal permutation representations of semidirect products of groups

Abstract We provide formulae for the minimal faithful permutation degree μ ⁢ ( G ) ${\mu(G)}$ of a group G that is a semidirect product of an elementary abelian p-group by a group of prime order q not equal to p. These formulae apply to the investigation of groups G with the property that there exists a nontrivial group H such that μ ⁢ ( G × H ) = μ ⁢ ( G ) ${\mu(G\times H)=\mu(G)}$ , in particular reproducing the seminal examples of Wright (1975) and Saunders (2010). Given an arbitrarily large group H that is a direct product of elementary abelian groups (with mixed primes), we construct a group G such that μ ⁢ ( G × H ) = μ ⁢ ( G ) ${\mu(G\times H)=\mu(G)}$ , yet G does not decompose nontrivially as a direct product.

On the $BP\langle n\rangle $-cohomology of elementary abelian $p$-groups

The structure of the BP-cohomology of elementary abelian p-groups is studied, obtaining a presentation expressed in terms of BP-cohomology and mod-p singular cohomology, using the Milnor derivations. The arguments are based on a result on multi-Koszul complexes which is related to Margolis's criterion for freeness of a graded module over an exterior algebra.

Association schemes in which the thin residue is an elementary abelian p-group of rank 2

In this article, we investigate the existence and schurity problem of association schemes whose thin residues are isomorphic to an elementary abelian p-group of rank 2.

A proof of Schwartz's conjecture about the eigenvalues of Lannes' T-functor

This paper gives an algebraic proof of a conjecture due to Lionel Schwartz which asserts that the linear operator induced by Lannes' T-functor on the Grothendieck group Knred generated by indecomposable summands of the mod p cohomology of a rank n elementary abelian p-group is diagonalizable over Q, with eigenvalues 1,p,…,pn and with multiplicities pn−pn−1,pn−1−pn−2,…,p−1,1, respectively. Using work of Harris and Shank, we first reduce this to an algebraic question involving the Grothendieck ring G0(Mn,p) of modules over the semigroup ring Fp[End(Fp⊕n)], showing that the induced action of T on Knred corresponds to the multiplication by an explicit element. In the second step, we establish the separability of the algebra C⊗G0(Mn,p), from which the diagonalizability and the computation of the eigenvalues and their multiplicities follow easily. The arguments use ingredients from the theory of Brauer characters of finite groups.

The extension algebra of some cohomological Mackey functors

Let k be a field of characteristic p. We construct a new inflation functor for cohomological Mackey functors for finite groups over k. Using this inflation functor, we give an explicit presentation of the graded algebra of self-extensions of the simple functor S1G, when p is odd and G is an elementary abelian p-group.

Group extensions and graphs

A classical result of Gaschütz affirms that given a finite A-generated group G and a prime p, there exists a group G# and an epimorphism φ:G#⟶G whose kernel is an elementary abelian p-group which is universal among all groups satisfying this property. This Gaschütz universal extension has also been described in the mathematical literature with the help of the Cayley graph. We give an elementary and self-contained proof of the fact that this description corresponds to the Gaschütz universal extension. Our proof depends on another elementary proof of the Nielsen–Schreier theorem, which states that a subgroup of a free group is free.

Nonsplit extensions of Abelian p-groups by L 2(p n ) and general theorems on extensions of finite groups

Let a group \(\tilde G\) be a nonsplit extension of an elementary Abelian p-group V by the group G = L 2(p n ) such that the action ofG on V is irreducible. In the present article, we classify (up to isomorphism) such groups \(\tilde G\) with p n ≠ 34.

Upper central series for elementary-abelian-over-cyclic regular wreath product p-groups

We compute the upper central series for the regular wreath product finite group C ≀ E where C is a cyclic p-group and E is an elementary abelian p-group for some prime p. The notion of a pattern subgroup enables us to describe the upper central series in a natural way.

RELATIVE RELATION MODULES OF FINITE ELEMENTARY ABELIAN p-GROUPS

Let E be a free product of a finite number of cyclic groups, and S a normal subgroup of E such that <TEX>$$E/S{\sim_=}G$$</TEX> is finite. For a prime p, <TEX>$\hat{S}=S/S^{\prime}S^p$</TEX> may be regarded as an <TEX>$F_pG$</TEX>-module via conjugation in E. The aim of this article is to prove that <TEX>$\hat{S}$</TEX> is decomposable into two indecomposable modules for finite elementary abelian p-groups G.

Universal deformation rings and fusion

We study in this paper the extent to which one can detect fusion in certain finite groups Γ from information about the universal deformation rings R(Γ,V) of absolutely irreducible FpΓ-modules V. The Γ we consider are extensions of either abelian or dihedral groups G of order prime to p by an elementary abelian p-group of rank 2.

On p-Schur Rings Over Abelian Groups of Order p 3

In the theory of Schur rings, it is known that every Schur ring over a cyclic p-group is Schurian. Recently, Spiga and Wang showed that every p-Schur ring over an elementary abelian p-group of rank 3 is Schurian. In this paper, we prove that every p-Schur ring over an abelian group of order p 3 is Schurian.

GENERIC JORDAN TYPE OF THE SYMMETRIC AND EXTERIOR POWERS

We prove a result relating the stable generic Jordan types of the symmetric and exterior powers of the Heller translations of a module for a finite elementary abelian p-group. In the case of the trivial module, the stable generic Jordan types of the symmetric and exterior powers of its Heller translations are completely described.

On the Error Detection Capability of One Check Digit

In this paper, we study a check digit system which is based on the use of elementary abelian p-groups of order pk. This paper is inspired by a recently introduced check digit system for hexadecimal numbers. By interpreting its check equation in terminology of matrix algebra, we generalize the idea to build systems over a group of order pk, while keeping the ability to detect all the: 1) single errors; 2) adjacent transpositions; 3) twin errors; 4) jump transpositions; and 5) jump twin errors. Besides, we consider two categories of jump errors: 1) t-jump transpositions and 2) t-jump twin errors, which include and further extend the double error types of 2)-5). In particular, we explore Rc, the maximum detection radius of the system on detecting these two kinds of generalized jump errors, and show that it is 2k-2 for p=2 and (pk-1)/2-1 for an odd prime p. Also, we show how to build such a system that detects all the single errors and these two kinds of double jump-errors within Rc.

K 2 of a Quotient Ring of ℤG

Let G be a finite abelian group and Γ the maximal ℤ-order of ℚG, an explicit description of Γ is given. When G is an elementary abelian p-group or a cyclic p-group, we give a lower bound for the order of K 2(ℤG/|G|Γ). For a finite abelian group G, we prove that K 2(ℤG/|G|Γ) is trivial if and only if |G| is square-free. Some discussions about the order of K 2(ℤG) are also given, especially for G a finite p-group when p is a regular prime.