scholarly journals A note on regular metabelian groups of prime-power order

1992 ◽  
Vol 46 (2) ◽  
pp. 343-346 ◽  
Author(s):  
M.F. Newman ◽  
Ming-Yao Xu
Keyword(s):  
Prime Power ◽  
Commutator Subgroup ◽  
Nilpotency Class ◽  
Prime Power Order ◽  
Metabelian Groups ◽  
Positive Integers

Let p be a prime and d, e positive integers. We prove that a regular d-generator metabelian p-group whose commutator subgroup has exponent pe has nilpotency class at most e(p – 2) + 1 unless e = 1, d > 2, p > 2 when the class can be p and these bounds are best possible.

scholarly journals Absolutely split metacyclic groups and weak metacirculants

2019 ◽  
Vol 18 (06) ◽  
pp. 1950117 ◽  
Author(s):  
Li Cui ◽  
Jin-Xin Zhou
Keyword(s):  
Automorphism Group ◽  
Prime Power ◽  
Necessary Condition ◽  
Prime Power Order ◽  
Metacyclic Groups ◽  
Vertex Transitive ◽  
Open Question ◽  
Positive Integers ◽  
The Relationship ◽  
Sufficient And Necessary Condition

Let [Formula: see text] be positive integers, and let [Formula: see text] be a split metacyclic group such that [Formula: see text]. We say that [Formula: see text] is absolutely split with respect to[Formula: see text] provided that for any [Formula: see text], if [Formula: see text], then there exists [Formula: see text] such that [Formula: see text] and [Formula: see text]. In this paper, we give a sufficient and necessary condition for the group [Formula: see text] being absolutely split. This generalizes a result of Sanming Zhou and the second author in [Weak metacirculants of odd prime power order, J. Comb. Theory A 155 (2018) 225–243]. We also use this result to investigate the relationship between metacirculants and weak metacirculants. Metacirculants were introduced by Alspach and Parsons in [Formula: see text] and have been a rich source of various topics since then. As a generalization of this class of graphs, Marušič and Šparl in 2008 introduced the so-called weak metacirculants. A graph is called a weak metacirculant if it has a vertex-transitive metacyclic automorphism group. In this paper, it is proved that a weak metacirculant of [Formula: see text]-power order is a metacirculant if and only if it has a vertex-transitive split metacyclic automorphism group. This provides a partial answer to an open question in the literature.

scholarly journals Central commutators

1984 ◽  
Vol 30 (1) ◽  
pp. 67-71
Author(s):  
A. Caranti ◽  
C.M. Scoppola
Keyword(s):  
Finite Groups ◽  
Prime Power ◽  
Commutator Subgroup ◽  
Prime Power Order

We give examples of finite groups of odd prime power order in which the commutators lying in the centre do not generate the intersection of the centre and the commutator subgroup.

scholarly journals Landau’s theorem on conjugacy classes for normal subgroups

2016 ◽  
Vol 26 (07) ◽  
pp. 1453-1466
Author(s):  
Antonio Beltrán ◽  
María José Felipe ◽  
Carmen Melchor
Keyword(s):  
Normal Subgroup ◽  
Finite Groups ◽  
Prime Power ◽  
Upper Bounds ◽  
Conjugacy Classes ◽  
Index Formula ◽  
Prime Power Order ◽  
Normal Subgroups ◽  
Small Index ◽  
Positive Integers

Landau’s theorem on conjugacy classes asserts that there are only finitely many finite groups, up to isomorphism, with exactly [Formula: see text] conjugacy classes for any positive integer [Formula: see text]. We show that, for any positive integers [Formula: see text] and [Formula: see text], there exist finitely many finite groups [Formula: see text], up to isomorphism, having a normal subgroup [Formula: see text] of index [Formula: see text] which contains exactly [Formula: see text] non-central [Formula: see text]-conjugacy classes. Upper bounds for the orders of [Formula: see text] and [Formula: see text] are obtained; we use these bounds to classify all finite groups with normal subgroups having a small index and few [Formula: see text]-classes. We also study the related problems when we consider only the set of [Formula: see text]-classes of prime-power order elements contained in a normal subgroup.

On metabelian groups of prime-power exponent

1968 ◽  
Vol 302 (1469) ◽  
pp. 237-242 ◽  
Keyword(s):  
Prime Power ◽  
Metabelian Group ◽  
Nilpotency Class ◽  
Power Exponent ◽  
Metabelian Groups

Meier-Wunderli has shown that every metabelian group of exponent p is nilpotent. Here we show that the subgroup generated by p k-1 th powers of elements in a metabelian group of exponent p k is nilpotent. We also obtain some information on the subgroup generated by p k-2 th powers. Finally we obtain a bound for the nilpotency class of n -generator metabelian groups of exponent p k .

scholarly journals Regular metabelian groups of prime-power order

1970 ◽  
Vol 3 (1) ◽  
pp. 49-54 ◽  
Author(s):  
R. J. Faudree
Keyword(s):  
Direct Product ◽  
Prime Power ◽  
Prime Power Order ◽  
Metabelian Groups ◽  
Finite Direct Product

Let H be a finite metabelian p-group which is nilpotent of class c. In this paper we will prove that for any prime p > 2 there exists a finite metacyclic p-group G which is nilpotent of class c such that H is isomorphic to a section of a finite direct product of G.

scholarly journals The energy of integral circulant graphs with prime power order

2011 ◽  
Vol 5 (1) ◽  
pp. 22-36 ◽  
Author(s):  
J.W. Sander ◽  
T. Sander
Keyword(s):  
Adjacency Matrix ◽  
Prime Power ◽  
Prime Power Order ◽  
Circulant Graph ◽  
Closed Formula ◽  
Circulant Graphs ◽  
Vertex Set

The energy of a graph is the sum of the moduli of the eigenvalues of its adjacency matrix. We study the energy of integral circulant graphs, also called gcd graphs. Such a graph can be characterized by its vertex count n and a set D of divisors of n such that its vertex set is Zn and its edge set is {{a,b} : a, b ? Zn; gcd(a-b, n)? D}. For an integral circulant graph on ps vertices, where p is a prime, we derive a closed formula for its energy in terms of n and D. Moreover, we study minimal and maximal energies for fixed ps and varying divisor sets D.

Space groups and groups of prime-power order II

1980 ◽  
Vol 35 (1) ◽  
pp. 203-209 ◽  
Author(s):  
H. Finken ◽  
J. Neub�ser ◽  
W. Plesken
Keyword(s):  
Prime Power ◽  
Prime Power Order ◽  
Space Groups

On S-Quasinormal and C-Normal Subgroups of Prime Power Order in Finite Groups

2011 ◽  
Vol 18 (04) ◽  
pp. 685-692
Author(s):  
Xuanli He ◽  
Shirong Li ◽  
Xiaochun Liu
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Prime Power ◽  
Maximal Subgroups ◽  
Prime Power Order ◽  
Normal Subgroups ◽  
A Finite Group

Let G be a finite group, p the smallest prime dividing the order of G, and P a Sylow p-subgroup of G with the smallest generator number d. Consider a set [Formula: see text] of maximal subgroups of P such that [Formula: see text]. It is shown that if every member [Formula: see text] of is either S-quasinormally embedded or C-normal in G, then G is p-nilpotent. As its applications, some further results are obtained.

scholarly journals Automorphism orbits of finite groups

1986 ◽  
Vol 40 (2) ◽  
pp. 253-260 ◽  
Author(s):  
Thomas J. Laffey ◽  
Desmond MacHale
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Finite Groups ◽  
Prime Power ◽  
Structure Theorem ◽  
Equivalence Classes ◽  
Prime Power Order ◽  
A Finite Group

AbstractLet G be a finite group and let Aut(G) be its automorphism group. Then G is called a k-orbit group if G has k orbits (equivalence classes) under the action of Aut(G). (For g, hG, we have g ~ h if ga = h for some Aut(G).) It is shown that if G is a k-orbit group, then kGp + 1, where p is the least prime dividing the order of G. The 3-orbit groups which are not of prime-power order are classified. It is shown that A5 is the only insoluble 4-orbit group, and a structure theorem is proved about soluble 4-orbit groups.

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