Metacyclic P-groups Research Articles

The Amit–Ashurst conjecture for finite metacyclic p-groups

The Amit conjecture about word maps on finite nilpotent groups has been shown to hold for certain classes of groups. The generalised Amit conjecture says that the probability of an element occurring in the image of a word map on a finite nilpotent group G is either 0, or at least 1/|G|. Noting the work of Ashurst, we name the generalised Amit conjecture the Amit–Ashurst conjecture and show that the Amit–Ashurst conjecture holds for finite p-groups with a cyclic maximal subgroup.

Projective characters of metacyclic p-groups

Let ω be a 2-cocycle of a metacyclic p-group G representing a non-trivial element of the Schur multiplier Then the number of ω-regular conjugacy classes of G, the subgroup consisting of the ω-regular elements in the center of G, the degree of each irreducible ω-character of G and a representation group H of G with M(H) trivial are all determined. Finally, for ω constructed from H, the projective character table of G corresponding to ω is found in the case that G is of positive type. Communicated by Mark Lewis

Conjugacy classes of maximal cyclic subgroups of metacyclic 𝑝-groups

Abstract In this paper, we set η ⁢ ( G ) \eta(G) to be the number of conjugacy classes of maximal cyclic subgroups of a finite group 𝐺. We compute η ⁢ ( G ) \eta(G) for all metacyclic 𝑝-groups. We show that if 𝐺 is a metacyclic 𝑝-group of order p n p^{n} that is not dihedral, generalized quaternion, or semi-dihedral, then η ⁢ ( G ) ≥ n - 2 \eta(G)\geq n-2 , and we determine when equality holds.

Some finite p-groups whose normal subgroups are characteristic

In this article, we classify some finite p-groups whose normal subgroups are characteristic. Particularly we restrict our attention to all finite p-groups of maximal class with an abelian maximal subgroup, finite 3-groups of maximal class, finite split metacyclic p-groups, minimal non-abelian p-groups and all p-groups of maximal class of order less than p 6.

The multiple holomorph of split metacyclic p-groups

Given any group G, the normalizer of the subgroup of left translations in the group of all permutations on G is called the holomorph, and the normalizer of in turn is called the multiple holomorph. The quotient has been computed for various families of groups G in the literature. In this paper, we shall supplement the existing results by considering finite split metacyclic p-groups G with p an odd prime. Our work gives new examples of groups G for which T(G) is not a 2-group.

Central units in some integral group rings

Let G be a finite group and be the integral group ring of G. We denote by the group of normalized units of that is, the units which have augmentation 1, and by the group of normalized central units. Many articles have been written describing the groups and for certain groups G. In this work, we will describe the group of normalized central units of some integral group rings by applying the idea presented in an article by Ferraz and Simón to a wider variety of groups, and we will study some examples of groups that can be treated with this method: metacyclic groups of type some metacyclic p-groups; some metabelian p-groups and some generalized dihedral groups.

Orderly Set Graph for Some Metacyclic p-Groups of Odd Prime

Orderly Set Graph for Some Metacyclic p-Groups of Odd Prime

On finite p-groups of supersoluble type

A finite p-group S is said to be of supersoluble type if every fusion system over S is supersoluble. The main aim of this paper is to characterise the finite p-groups of supersoluble type. Abelian and metacyclic p-groups of supersoluble type are completely described. Furthermore, we show that the Sylow p-subgroups of supersoluble type of a finite simple group must be cyclic.

Counting pattern-avoiding integer partitions

A partition $$\alpha $$ is said to contain another partition (or pattern) $$\mu $$ if the Ferrers board for $$\mu $$ is attainable from $$\alpha $$ under removal of rows and columns. We say $$\alpha $$ avoids $$\mu $$ if it does not contain $$\mu $$ . In this paper we count the number of partitions of n avoiding a fixed pattern $$\mu $$ , in terms of generating functions and their asymptotic growth rates. We find that the generating function for this count is rational whenever $$\mu $$ is (rook equivalent to) a partition in which any two part sizes differ by at least two. In doing so, we find a surprising connection to metacyclic p-groups. We further obtain asymptotics for the number of partitions of n avoiding a pattern $$\mu $$ . Using these asymptotics we conclude that the generating function for $$\mu $$ is not algebraic whenever $$\mu $$ is rook equivalent to a partition with distinct parts whose first two parts are positive and differ by 1.

Edge-transitive bi-Cayley graphs

A graph Γ admitting a group H of automorphisms acting semi-regularly on the vertices with exactly two orbits is called a bi-Cayley graph over H. This generalisation of a Cayley graph gives a class of graphs that includes many important examples such as the Petersen graph, the Gray graph and the Hoffman-Singleton graph. A bi-Cayley graph Γ over a group H is called normal if H is normal in the full automorphism group of Γ, and almost-normal edge-transitive if the normaliser of H in the full automorphism group of Γ is transitive on the edges of Γ. In this paper, we give a characterisation of almost-normal edge-transitive bi-Cayley graphs, and in particular, we give a detailed description of 2-arc-transitive normal bi-Cayley graphs. Using this, we investigate three classes of bi-Cayley graphs, namely those over abelian groups, dihedral groups and metacyclic p-groups. We find that under certain conditions, ‘almost-normal edge-transitive’ is the same as ‘normal’ for graphs in these three classes. As a by-product, we obtain a complete classification of all connected trivalent edge-transitive graphs of girth at most 6. Also we answer some open questions from the literature about these and related classes of graphs, posed by Li (in Proc. Amer. Math. Soc. 133 (2005)), Marušič and Potočnik (in Eur. J. Comb. 22 (2001)) and Marušič and Šparl (in J. Algebr. Comb. 28 (2008)), and we correct a major error in a recent paper by Li, Song and Wang (in J. Comb. Theory Ser. A 120 (2013)).

The Subgroups of Finite Metacyclic Groups

In this paper, the author characterizes the subgroups of a finite metacyclic group K by building a one to one correspondence between certain 3-tuples (k,l,β) ∈ ℕ3 and all the subgroups of K. The results are applied to compute some subgroups of K as well as to study the structure and the number of p-subgroups of K, where p is a fixed prime number. In addition, the author gets a factorization of K, and then studies the metacyclic p-groups, gives a different classification, and describes the characteristic subgroups of a given metacyclic p-group when p ≥ 3. A “reciprocity” relation on enumeration of subgroups of a metacyclic group is also given.

Abdollahi’s Conjectures and a Class of Prime Power Group

In this paper, we give necessary and sufficient conditions for metacyclic p-groups H and K to have isomorphic non-commuting graphs.

On the Chermak–Delgado lattices of split metacyclic p-groups

The Chermak–Delgado measure of a subgroup H of a finite group G is defined as mG(H) = |H||CG(H)|. The subgroups with maximal Chermak–Delgado measure form a poset and corresponding lattice, known as the CD-lattice of G. We describe the symmetric nature of CD-lattices in general, and use information about centrally large subgroups to determine the CD-lattices of split metacyclic p-groups in particular. We also describe a rank-symmetric sublattice of the CD-lattice of split metacyclic p-groups.

Computing Whitehead groups using genetic bases

We combine results about Whitehead groups of finite groups with results about genetic bases of finite p-groups to compute the Whitehead groups of some metacyclic p-groups. Let Cpn denote a cyclic group of order pn for p an odd prime and n a positive integer. We present a conjecture for the torsion part of the Whitehead group of Cpn×Cpn and we show that torsion part of the Whitehead group of the semidirect product of Cpn−1 by Cp is isomorphic to Cp(n−2)(p−1). The techniques we use can be applied to any abelian p-group and to many other p-groups for p odd.

CHARACTER TABLES OF METACYCLIC GROUPS

AbstractWe show that two metacyclic groups of the following types are isomorphic if they have the same character tables: (i) split metacyclic groups, (ii) the metacyclic p-groups and (iii) the metacyclic {p, q}-groups, where p, q are odd primes.

On some metacycli cp-groups and their nonabelian tensor square

On some metacycli cp-groups and their nonabelian tensor square

Maximal abelian and minimal nonabelian subgroups of some finite two-generator p-groups especially metacyclic

We study the subgroup structure of some two-generator p-groups and apply the obtained results to metacyclic p-groups. For metacyclic p-groups G, p > 2, we do the following: (a) compute the number of nonabelian subgroups with given derived subgroup, show that (ii) minimal nonabelian subgroups have equal order, (c) maximal abelian subgroups have equal order, (d) every maximal abelian subgroup is contained in a minimal nonabelian subgroup and all maximal subgroups of any minimal nonabelian subgroup are maximal abelian in G. We prove the same results for metacyclic 2-groups (e) with abelian subgroup of index p, (f) without epimorphic image ≅ D8. The metacyclic p-groups containing (g) a minimal nonabelian subgroup of order p4, (h) a maximal abelian subgroup of order p3 are classified. We also classify the metacyclic p-groups, p > 2, all of whose minimal nonabelian subgroups have equal exponent. It appears that, with few exceptions, a metacyclic p-group has a chief series all of whose members are characteristic.

Computing the Exterior Center of Metacyclic p-groups of Nilpotency Class at Least Three

A group G is metacyclic if it contains a cyclic normal subgroup K such that G/K is also cyclic. Metacyclic p-groups classified by different authors. A group is called capable if it is a central factor group. The purpose of this study is to compute the exterior center of finite non-abelian metacyclic p-groups, p is an odd prime, for some small order groups using Groups, Algorithms and Programming (GAP) software. We also determine which of these groups are capable.

Conjugacy classes and commuting probability in finite metacyclic p-groups

Let G be a finite non-abelian metacyclic p-group where p is any prime. We compute the exact number of conjugacy classes and the commutativity degree of G. In particular, we describe the number of conjugacy classes both in the split and non-split case.

On automorphisms of order p of metacyclic p-groups without cyclic subgroups of index p

On automorphisms of order p of metacyclic p-groups without cyclic subgroups of index p