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 Lattice embeddings of abelian prime power groups

1997 ◽  
Vol 62 (2) ◽  
pp. 259-278 ◽  
Author(s):  
Roland Schmidt
Keyword(s):  
Direct Product ◽  
Prime Power ◽  
Abelian Groups ◽  
Subgroup Lattice ◽  
Prime Power Order

AbstractWe solve the following problem which was posed by Barnes in 1962. For which abelian groups G and H of the same prime power order is it possible to embed the subgroup lattice of G in that of H? It follows from Barnes' results and a theorem of Herrmann and Huhn that if there exists such an embedding and G contains three independent elements of order p2, then G and H are isomorphic. This reduces the problem to the case that G is the direct product of cyclic p-groups only two of which have order larger than p. We determine all groups H for which the desired embedding exists.

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.

On Simply Transitive Primitive Permutation Groups

1969 ◽  
Vol 21 ◽  
pp. 1062-1068 ◽  
Author(s):  
R. D. Bercov
Keyword(s):  
Abelian Group ◽  
Direct Product ◽  
Prime Power ◽  
Direct Decomposition ◽  
Permutation Groups ◽  
Prime Power Order ◽  
Schur Ring ◽  
Abelian Subgroups ◽  
Prime 2

In (1) we considered finite primitive permutation groups G with regular abelian subgroups H satisfying the following hypothesis:(*) H = A × B × C, where A is cyclic of prime power order pα ≠ 4, B has exponent pβ < pα, and C has order prime to p.We remark that an abelian group fails to satisfy (*) (apart from the minor exception associated with the prime 2) if and only if it is the direct product of two subgroups of the same exponent.We showed in (1) that such a group G is doubly transitive unless it is the direct product of two or more subgroups each of the same order greater than 2. This was done by showing that (in the terminology of (3)) the existence of a non-trivial primitive Schur ring over H implies such a direct decomposition of H.

scholarly journals Isomorphisms of Direct Products of Cyclic Groups of Prime Power Order

2013 ◽  
Vol 21 (3) ◽  
pp. 207-211
Author(s):  
Hiroshi Yamazaki ◽  
Hiroyuki Okazaki ◽  
Kazuhisa Nakasho ◽  
Yasunari Shidama
Keyword(s):  
Direct Product ◽  
Cyclic Group ◽  
Prime Power ◽  
Prime Power Order ◽  
Direct Products ◽  
Cyclic Groups

Summary In this paper we formalized some theorems concerning the cyclic groups of prime power order. We formalize that every commutative cyclic group of prime power order is isomorphic to a direct product of family of cyclic groups [1], [18].

scholarly journals On finite loops whose inner mapping groups are Abelian II

2005 ◽  
Vol 71 (3) ◽  
pp. 487-492
Author(s):  
Markku Niemenmaa
Keyword(s):  
Abelian Group ◽  
Direct Product ◽  
Prime Power ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Prime Power Order ◽  
Finite Abelian Groups ◽  
Cyclic Groups ◽  
Inner Mapping Group ◽  
Mapping Group

If the inner mapping group of a loop is a finite Abelian group, then the loop is centrally nilpotent. We first investigate the structure of those finite Abelian groups which are not isomorphic to inner mapping groups of loops and after this we show that if the inner mapping group of a loop is isomorphic to the direct product of two cyclic groups of the same odd prime power order pn, then our loop is centrally nilpotent of class at most n + 1.

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.

Sign in / Sign up

Export Citation Format

Share Document