An Elementary Classification of Finite Metacyclic p-Groups of Class at Least Three

Beuerle, James R.

doi:10.1142/s1005386705000519

Cited by 23 publications

(23 citation statements)

References 6 publications

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“…In 2005, Beuerle [9] gives several classifications of finite metacyclic p-groups of class at least three. In this paper, we used the classification of metacyclic p-groups of class at least three for odd prime, as given in the following theorem.…”

Section: G Gh X G Hx G H G X X X X X Gmentioning

confidence: 99%

The probability that an element of a metacyclic 3-Group of negative type fixes a set and its orbit graph

Zamri

Sarmin²,

Omer

et al. 2016

AIP Conference Proceedings

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Section: G Gh X G Hx G H G X X X X X Gmentioning

confidence: 99%

The probability that an element of a metacyclic 3-Group of negative type fixes a set and its orbit graph

Zamri

Sarmin²,

Omer

et al. 2016

AIP Conference Proceedings

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“…Then Case I covers the types (1), (3), (4), (5), and (6), Case II covers the forms (2), (7), and (8), and Case III covers the forms (9), (10), and (11). In what follows, we will write 'metacyclic p-group' to mean metacyclic p-groups of all cases.…”

Section: Introductionmentioning

confidence: 99%

Some results on the non-commuting graph of a finite group

Moradipour¹,

Ilangovanb²,

Rashidc³

2019

ScienceAsia

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Let G be a metacyclic p-group, and let Z(G) be its center. The non-commuting graph Γ G of a metacyclic pgroup G is defined as the graph whose vertex set is G−Z(G), and two distinct vertices x and y are connected by an edge if and only if the commutator of x and y is not the identity. In this paper, we give some graph theoretical properties of the non-commuting graph Γ G . Particularly, we investigate planarity, completeness, clique number and chromatic number of such graph. Also, we prove that if G 1 and G 2 are isoclinic metacyclic p-groups, then their associated graphs are isomorphic.

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“…Beuerle classified the metacyclic p-groups as being either metacyclic p-groups of nilpotency of class two or metacyclic p-groups of nilpotency of class at least three 5 . Based on Beuerle's classification, the metacyclic p-groups of nilpotency class two are partitioned into two families of non-isomorphic p-groups (types (i) and (ii) in the following list).…”

Section: Introductionmentioning

confidence: 99%

“…Based on Beuerle's classification, the metacyclic p-groups of nilpotency class two are partitioned into two families of non-isomorphic p-groups (types (i) and (ii) in the following list). The metacyclic 2-groups of negative type of class at least three are partitioned into eight families 5 . Types (iii)-(x) in the following list are some of the negative types that are considered here.…”

Section: Introductionmentioning

confidence: 99%

Some applications of metacyclic 2-groups of negative type

Omer

Sarmin²,

Erfanian

2016

ScienceAsia

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ABSTRACT:The probability that two random elements commute in a finite group G is the quotient of the number of commuting elements and |G| 2 . Consider a set S consisting of all subsets of commuting elements of G of size two that are in the form (a, b) where a and b commute and lcm(|a|, |b|) = 2. The probability that a group element fixes S is the number of orbits under the group action on S divided by |S|. In this paper, the probability that a group element fixes a set S under regular action is found for metacyclic 2-groups of negative type of nilpotency class two and of class at least three. The results obtained from the sizes of the orbits are then applied to the generalized conjugacy class graph.

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An Elementary Classification of Finite Metacyclic p-Groups of Class at Least Three

Abstract: In this paper, we classify metacyclic p-groups of nilpotency class at least three up to isomorphism by giving a canonical presentation for each isomorphism class.

Cited by 23 publications

References 6 publications

The probability that an element of a metacyclic 3-Group of negative type fixes a set and its orbit graph

The probability that an element of a metacyclic 3-Group of negative type fixes a set and its orbit graph

Some results on the non-commuting graph of a finite group

Some applications of metacyclic 2-groups of negative type

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