Extraspecial P-group Research Articles

On the twisted group ring isomorphism problem for a class of groups

The twisted group ring isomorphism problem (TGRIP) is a variation of the classical group ring isomorphism problem. It asks whether the ring structure of the twisted group ring determines the group up to isomorphism. In this article, we study the TGRIP for direct product and central product of groups. We provide some criteria to answer the TGRIP for groups by answering the TGRIP for certain associated quotients. As an application of these results, we provide several examples. Finally, we answer the TGRIP for extra-special p-groups, and for all but five groups of order p5, where p≥5 is prime.

Word automatic groups of nilpotency class 2

We consider word automaticity for groups that are nilpotent of class 2 and have exponent a prime p. We show that the infinitely generated free group in this variety is not word automatic. In contrast, the infinite extra-special p-group Ep is word automatic, as well as an intermediate group Hp which has an infinite centre. In the last section we introduce a method for showing automaticity of central extensions of abelian groups via co-cycles.

Structure of blocks with normal defect and abelian inertial quotient

Abstract Let k be an algebraically closed field of prime characteristic p. Let $kGe$ be a block of a group algebra of a finite group G, with normal defect group P and abelian $p'$ inertial quotient L. Then we show that $kGe$ is a matrix algebra over a quantised version of the group algebra of a semidirect product of P with a certain subgroup of L. To do this, we first examine the associated graded algebra, using a Jennings–Quillen style theorem. As an example, we calculate the associated graded of the basic algebra of the nonprincipal block in the case of a semidirect product of an extraspecial p-group P of exponent p and order $p^3$ with a quaternion group of order eight with the centre acting trivially. In the case of $p=3$ , we give explicit generators and relations for the basic algebra as a quantised version of $kP$ . As a second example, we give explicit generators and relations in the case of a group of shape $2^{1+4}:3^{1+2}$ in characteristic two.

Extra-special p-groups as groups of automorphisms

Let p be a prime and E an extra-special p-group of exponent p and order Suppose that E acts by automorphisms on a finite -group G. We prove that if is nilpotent for any then is nilpotent. If, for some integer d such that the dth derived group of is nilpotent for any then the dth derived group is nilpotent. We also prove similar results for Lie algebras.

Projective characters of extra-special p-groups

Let G be an extra-special p-group of order for The aim of this paper is to investigate what possible combinations of degrees (counting multiplicities) of irreducible α-characters of G can occur for in the Schur multiplier and then to determine the numbers of that give the same combinations. Solutions are given when n = 2 and in general for Writing G for n > 2 as the central product of two extra-special p-groups E and K, partial solutions are obtained for p odd when (a) every element of E is α-regular for all and (b) and the subgroup of M(G) consisting of with trivial is considered instead.

A note on the power graphs of finite nilpotent groups

The power graph P(G) of a group G is the graph with vertex set G and two distinct vertices are adjacent if one is a power of the other. Two finite groups are said to be conformal, if they contain the same number of elements of each order. Let Y be a family of all non-isomorphic odd order finite nilpotent groups of class two or p-groups of class less than p. In this paper, we prove that the power graph of each group in Y is isomorphic to the power graph of an abelian group and two groups in Y have isomorphic power graphs if they are conformal. We determine the number of maximal cyclic subgroups of a generalized extraspecial p-group (p odd) by determining the power graph of this group. We also determine the power graph of a p-group of order p4 (p odd).

Self-complementary Cayley Graphs of Extraspecial p-groups

This paper constructs several families of self-complementary Cayley graphs of extraspecial p-groups, where p is a prime and congruent to 1 modulo 4.

The Whitehead group of (almost) extra-special p-groups with p odd

Let p be an odd prime number. We describe the Whitehead group of all extra-special and almost extra-special p-groups. For this we compute, for any finite p-group P, the subgroup Cl1(ZP) of SK1(ZP), in terms of a genetic basis of P. We also introduce a deflation map Cl1(ZP)→Cl1(Z(P/N)), for a normal subgroup N of P, and show that it is always surjective. Along the way, we give a new proof of the result describing the structure of SK1(ZP), when P is an elementary abelian p-group.

Spherical posets from commuting elements

Abstract In this paper, we study the homotopy type of the partially ordered set of left cosets of abelian subgroups in an extraspecial p-group. We prove that the universal cover of its nerve is homotopy equivalent to a wedge of r-spheres where 2 ⁢ r ≥ 4 {2r\geq 4} is the rank of its Frattini quotient. This determines the homotopy type of the universal cover of the classifying space of transitionally commutative bundles as introduced in [2].

On the Brauer–Picard groups of fusion categories

We develop methods of computation of the Brauer–Picard groups of fusion categories and apply them to compute such groups for several classes of fusion categories of prime power dimension: representation categories of elementary abelian groups with twisted associativity, extra special p-groups, and the Kac–Paljutkin Hopf algebra. We conclude that many finite groups of Lie type occur as composition factors of the Brauer–Picard groups of pointed fusion categories.

Finite p-groups in which each absolute central automorphism is elementary abelian

In this paper we give necessary and sufficient conditions on a p-group G for each absolute central automorphism of G to be elementary abelian. Also we classify the absolute centre of extraspecial p-groups and show that the absolute central automorphisms of these groups are elementary abelian.

Regular Orbits of Extra-special Groups

In this paper, for odd primes p we find a recursive formula for the number of regular orbits of an extra-special p-group of exponent p2 acting on a faithful irreducible module over a finite field. This complements an earlier result of Foulser for extra-special groups of order p.

Representations of the double Burnside algebra and cohomology of the extraspecial p-group II

Let E be the extraspecial p-group of order p3 and exponent p where p is an odd prime. We determine the modp cohomology H⁎(X,Fp) of a summand X in the stable splitting of p-completed classifying space BE. In the previous paper (Hida and Yagita (2014) [7]), we determined this cohomology modulo nilpotence. In this paper, we consider the whole of the cohomology. Moreover, we consider the stable splittings of BG for some finite groups with Sylow p-subgroup E related to the three dimensional linear group L3(p).

The MOR Cryptosystem And Extra-Special p-Groups

This paper studies the MOR cryptosystem, using the automorphism group of the extra- special p-group of exponent p, for an odd prime p. Similar results can be obtained for extra-special p-groups of exponent p2 and for the even prime.

Representations of the double Burnside algebra and cohomology of the extraspecial p-group

Let E be the extraspecial p-group of order p3 and exponent p where p is an odd prime. We determine the mod p cohomology of summands in the stable splitting of p-completed classifying space BE modulo nilpotence. It is well known that indecomposable summands in the complete stable splitting correspond to simple modules for the mod p double Burnside algebra. We shall use representation theory of the double Burnside algebra and the theory of biset functors.

A NOTE ON THE RELATIVE COMMUTATIVITY DEGREE OF FINITE GROUPS

The purpose of this paper is to give a relation between the notion of the commutativity degree of a finite group G (denoted by d(G)) and that of isoclinism between G and an extra special p-group, where p is the smallest prime number dividing |G|. Moreover, some improvements of the results on the relative commutativity degree and relative n th nilpotency degree of a subgroup of finite groups given in [A. Erfanian, R. Rezaei and P. Lescot, On the relative commutativity degree of a subgroup of a finite group, Comm. Algebra35 (2007) 4183–4197] are also stated in this paper.

Some Properties of Certain Subgroups of Tensor Squares of p-Groups

Let G be a finite p-group of order p n and ∇(G) be the subgroup of the tensor square of G generated by all symbols x ⊗ x, for all x in G. In the present article, we construct an upper bound for the order of ∇(G) and any extra special p-group. It is also shown that ∇(G) ≅ ∇(G/G′). Using our result, we obtain the explicit structure of the tensor square of G and π3 SK(G, 1). Finally, the structure of G will be characterized when the bound is attained.

A Bound on the Order of Non-Abelian Tensor Square of a Prime-Power Group

This article improves on an upper bound for the order of the non-abelian tensor square of a finite p-group G given in [3]. In particular, applying this for finite p-groups of order p n with factor group G/G′ of order p m , the bound p nm attains if and only if G is elementary abelian of rank n, quaternion group of order 8, or extra special p-group of order p 3 with odd exponent p.

Regular orbits of finite primitive solvable groups, II

Suppose that a finite solvable group G acts faithfully, irreducibly and quasi-primitively on a finite vector space V. Then G has a uniquely determined normal subgroup E which is a direct product of extraspecial p-groups for various p and we denote e = | E / Z ( E ) | . We prove that when e = 5 , 6 , 7 , G will have regular orbits on V. This result, combined with a previous result of the author, will give a complete list of e such that G will have regular orbits on V.

On the automorphism group of a generalized extraspecial p-group

Abstract generalized extraspecial p-groups symplectic groups central product automorphisms × Update Abstract English Chinese Update