P-group Of Order Research Articles

CONJUGACY CLASSES OF MAXIMAL CYCLIC SUBGROUPS AND NILPOTENCE CLASS OF -GROUPS

AbstractLet $\eta (G)$ be the number of conjugacy classes of maximal cyclic subgroups of G. We prove that if G is a p-group of order $p^n$ and nilpotence class l, then $\eta (G)$ is bounded below by a linear function in $n/l$ .

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.

Double bordered constructions of self-dual codes from group rings over Frobenius rings

In this work, we describe a double bordered construction of self-dual codes from group rings. We show that this construction is effective for groups of order 2p where p is odd, over the rings mathbb {F}_{2}+umathbb {F}_{2} and mathbb {F}_{4}+umathbb {F}_{4}. We demonstrate the importance of this new construction by finding many new binary self-dual codes of lengths 64, 68 and 80; the new codes and their corresponding weight enumerators are listed in several tables.

On the capability of finite p-groups with derived subgroup of order p

A group is said to be capable if it is a central factor group. Let denotes the class of finite p-groups with derived subgroup of order p and central factor group of order p2. In this paper for groups in , we compute the various homological functors, among them the nonabelian tensor square and the Schur multiplier. Furthermore, the epicenters of all these groups are determined to give a complete classification of finite capable p-groups with derived subgroup of order p.

2-Arc-Transitive Regular Covers of Kn,n Having the Covering Transformation Group ℤp2

This paper contributes to the classification of finite 2-arc-transitive graphs. In [12], all the regular covers of complete bipartite graphs Kn,n were classified, whose covering transformation group is cyclic and whose fibre-preserving automorphism group acts 2-arc-transitively. In this paper, a further classification is achieved for all the regular covers of Kn,n, whose covering transformation group is elementary abelian group of order p2 and whose fibre-preserving automorphism group acts 2-arc-transitively. As a result, two new infinite families of 2-arc-transitive graphs are found. Moveover, it will be explained that it seems to be infeasible to classify all such covers when the covering transformation group is an elementary abelian group of order pk for an arbitrary integer k.

Finite dimensional Nichols algebras over certain p-groups

ABSTRACTWe investigate pointed Hopf algebras over finite nilpotent groups of odd order, with nilpotency class 2. For such a group G, we show that if its commutator subgroup coincides with its center, then there exists no non-trivial finite-dimensional pointed Hopf algebra with kG as its coradical. We apply these results to non-abelian groups of order p3, p4 and p5, and list all the pointed Hopf algebras of order p6, whose group of grouplikes is non-abelian.

Finite Metabelian Group Algebras

Given a finite metabelian group G, whose central quotient is abelian (not cyclic) group of order p2, p odd prime, the objective of this paper is to obtain a complete algebraic structure of semisimple group algebra Fq[G] in terms of primitive central idempotents, Wedderburn decomposition and the automorphism group.

Some Characterizations of a Normal Subgroup of a Group and Isotopic Classes of Transversals

Let G be a group and H be a subgroup of G which is either finite or of finite index in G. In this paper, we give some characterizations for the normality of H in G. As a consequence we get a very short and elementary proof of the main theorem of a paper of Lal and Shukla, which avoids the use of the classification of finite simple groups. Further, we study the isotopy between the transversals in some groups and determine the number of isotopy classes of transversals of a subgroup of order 2 in D2p, the dihedral group of order 2p, where p is an odd prime and the isotopism classes are formed with respect to induced right loop structures.

A family of non-Schurian p-Schur rings over groups of order p3${p^{3}}$

Abstract Recently, it was proved that every commutative p-Schur ring over a group of order p 3 ${p^{3}}$ is Schurian. In this article, we consider the Schurity problem of non-commutative p-Schur rings over groups of order p 3 ${p^{3}}$ . In particular, we give a family of non-Schurian p-Schur rings over groups of order p 3 ${p^{3}}$ .

Units of ℤ(Cp × C2) and ℤ(Cp × C2 × C2)

The goal of this article is to describe a multiplicatively independent set, which generates the group of units of the integral group ring ℤG, where G is either the cyclic group of order 2p or C2 × C2 × Cp, for a prime number p that satisfies some suitable conditions that will be specified later.

Dihedral Groups of Order 2p of Automorphisms of Compact Riemann Surfaces of Genus p – 1

AbstractIn this paper we prove that there is only one conjugacy class of dihedral group of order 2p in the 2(p – 1) × 2(p – 1) integral symplectic group that can be realized by an analytic automorphism group of compact connected Riemann surfaces of genus p – 1. A pair of representative generators of the realizable class is also given.

Some remarks on subgroups of maximal class in a finite p-group

The following characterizations of p-groups of maximal class are proved: (a) If a p-group of order > pp+2 contains a subgroup of maximal class and index p, then G possesses at most one normal subgroup of order pp and exponent p. (b) If the center of any nonabelian epimorphic image of a nonabelian two-generator p-group G is cyclic, then either G ≅ M pn or G is of maximal class. (c) An 𝒜n-group G, n > 1, is of maximal class ⇔ all its 𝒜2-subgroups of minimal order are of maximal class. (iv) If all factors of the lower central series of a nonabelian two-generator p-group are cyclic, then it is either of maximal class or ≅ M pn. (v) If a nonabelian p-group G is such that any s pairwise non-commuting elements generate a group of maximal class, where s is the fixed member of the set {3, …, p + 1} and p > 2 if s ≠ p + 1, then G is also of maximal class. We also study the noncyclic p-groups containing only one normal subgroup of a given order.

Characterization of Finite p-Groups by Their Non-Abelian Tensor Square

Let G be a finite p-group of order p n with |G′| =p c and G ⊗ G be the non-abelian tensor square of G. In 1991, Rocco proved that |G ⊗ G| =p n(n−c)−l for some integer l ≥ 0. In this article we characterize all finite p-groups, for which |G′| =p and 0 ≤ l ≤ 10.

A Note on Beauville p-Groups

We examine which p-groups of order ⩽p 6 are Beauville. We completely classify them for groups of order ⩽p 4. We also show that the proportion of 2-generated groups of order p 5 that are Beauville tends to 1 as p tends to infinity; this is not true, however, for groups of order p 6. For each prime p we determine the smallest nonabelian Beauville p-group.

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.

Arc-transitive Dihedrants of Odd Prime-power Order

Let G be a finite group with identity element 1, and S be a subset of G such that $${1 \notin S}$$ and S = S ?1. The Cayley graph Cay(G, S) has vertex set G, and x, y in G are adjacent if and only if $${xy^{-1} \in S}$$ . In this paper we classify the connected, arc-transitive Cayley graphs $${{\rm Cay}(D_{2p^n}, S),}$$ where $${D_{2p^n}}$$ is the dihedral group of order 2p n , p is an odd prime.

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.

Projective Character Degree Patterns of p-Groups for p Odd

In Higgs and Humphreys [7], the 2-groups that realize all their possible projective character degree patterns were classified. The p-groups of order p n , for n ≤ 4 and p odd, that satisfy this property are also all known. In this article, we will complete the classification of all p-groups that have this property. This will be achieved by showing that a p-group of order p n , for n ≥ 5 and p odd, cannot realize all its possible projective character degree patterns.

On finite p-groups with few nonabelian subgroups of order p p and exponent p

Let G be a finite p-group. If p = 2, then a nonabelian group G = Ω1(G) is generated by dihedral subgroups of order 8. If p > 2 and a nonabelian group G = Ω1(G) has no subgroup isomorphic to \({\Sigma _{{p^2}}}\), a Sylow p-subgroup of the symmetric group of degree p2, then it is generated by nonabelian subgroups of order p3 and exponent p. If p > 2 and the irregular p-group G has 2, and G is a p-group of order > pp+1, then the number of subgroups ≅ Σ\({\Sigma _{{p^2}}}\) in G is a multiple of p.

A note on p-groups of order ≤ p 4

In [1], we defined c(G), q(G) and p(G). In this paper we will show that if G is a p-group, where p is an odd prime and |G| ≤ p4, then c(G) = q(G) = p(G). However, the question of whether or not there is a p-group G with strict inequality c(G) = q(G) < p(G) is still open.