THE STRONG SYMMETRIC GENUS OF DIRECT PRODUCTS

2011 ◽  
Vol 10 (05) ◽  
pp. 901-914 ◽  
Author(s):  
COY L. MAY

Let G be a finite group. The strong symmetric genus σ0(G) is the minimum genus of any Riemann surface on which G acts faithfully and preserving orientation. Assume that G is non-abelian and generated by two elements, one of which is an involution, and that n is relatively prime to |G|. Our first main result is the determination of the strong symmetric genus of the direct product Zn ×G in terms of n, |G|, and a parameter associated with the group G. We obtain a variety of genus formulas. Finally, we apply these results to prove that for each integer g ≥ 2, there are at least four groups of strong symmetric genus g.

Author(s):  
D. R. Taunt

It is well known that a characteristically-simple finite group, that is, a group having no characteristic subgroup other than itself and the identity subgroup, must be either simple or the direct product of a number of isomorphic simple groups. It was suggested to the author by Prof. Hall that finite groups possessing exactly one proper characteristic subgroup would repay attention. We shall call a finite group having a unique proper characteristic subgroup a ‘UCS group’. In the present paper we first give some results on direct products of isomorphic UCS groups, and then we consider in more detail one of the types of UCS groups which can exist, that consisting of groups whose orders are divisible by exactly two distinct primes.


1988 ◽  
Vol 31 (3) ◽  
pp. 469-474
Author(s):  
Robert W. van der Waall

Let K be a field, G a finite group, V a (right) KG-module. If H is a subgroup of G, then, restricting the action of G on V to H, V is also a KH-module. Notation: VH.Suppose N is a normal subgroup of G. The KN-module VN is not irreducible in general, even when V is irreducible as KG-module. A part of the well-known theorem of A. H. Clifford [1, V.17.3] yields the following.


2015 ◽  
Vol 67 (4) ◽  
pp. 848-869 ◽  
Author(s):  
Bernhard Köck ◽  
Joseph Tait

AbstractGiven a faithful action of a finite groupGon an algebraic curveXof genusgX≥ 2, we giveexplicit criteria for the induced action ofGon the Riemann–Roch spaceH0(X,OX(D)) to be faithful,whereDis aG-invariant divisor on X of degree at least 2gX− 2. This leads to a concise answer to the question of when the action ofGon the spaceH0(X,Ωx⊗m) of global holomorphic polydifferentials of order m is faithful. IfXis hyperelliptic, we provide an explicit basis of H0(X,Ωx⊗m). Finally, we giveapplications in deformation theory and in coding theory and discuss the analogous problem for theaction ofGon the first homologyH1(X,ℤ/mℤ) ifXis a Riemann surface.


1985 ◽  
Vol 63 (8) ◽  
pp. 1061-1064 ◽  
Author(s):  
B. Lulek ◽  
T. Lulek ◽  
R. Chatterjee ◽  
J. Biel

The method of evaluation of Clebsch–Gordan coefficients to calculate the direct product of transitive representations for a finite group is developed. This procedure is based on the Mackey theorem and on the canonical realization of transitive representations. It is shown that the Clebsch–Gordan decomposition is associated with a natural interpretation of an orbit of a resultant transitive representation and it is analogous to a fibre bundle with the fibration being determined by constituent transitive representations.


Author(s):  
D. H. McLain ◽  
P. Hall

1. If P is any property of groups, then we say that a group G is ‘locally P’ if every finitely generated subgroup of G satisfies P. In this paper we shall be chiefly concerned with the case when P is the property of being nilpotent, and will examine some properties of nilpotent groups which also hold for locally nilpotent groups. Examples of locally nilpotent groups are the locally finite p-groups (groups such that every finite subset is contained in a finite group of order a power of the prime p); indeed, every periodic locally nilpotent group is the direct product of locally finite p-groups.


2018 ◽  
Vol 61 (2) ◽  
pp. 381-423
Author(s):  
JÜRGEN MÜLLER ◽  
SIDDHARTHA SARKAR

AbstractThe genus spectrum of a finite group G is the set of all g such that G acts faithfully on a compact Riemann surface of genus g. It is an open problem to find a general description of the genus spectrum of the groups in interesting classes, such as the Abelian p-groups. Motivated by earlier work of Talu for odd primes, we develop a general combinatorial method, for arbitrary primes, to obtain a structured description of the so-called reduced genus spectrum of Abelian p-groups, including the reduced minimum genus. In particular, we determine the complete genus spectrum for a large subclass, namely, those having ‘large’ defining invariants. With our method we construct infinitely many counterexamples to a conjecture of Talu, which states that an Abelian p-group is recoverable from its genus spectrum. Finally, we give a series of examples of our method, in the course of which we prove, for example, that almost all elementary Abelian p-groups are uniquely determined by their minimum genus, and that almost all Abelian p-groups of exponent p2 are uniquely determined by their minimum genus and Kulkarni invariant.


1968 ◽  
Vol 20 ◽  
pp. 1300-1307 ◽  
Author(s):  
Fletcher Gross

A finite group G is said to be a fixed-point-free-group (an FPF-group) if there exists an automorphism a which fixes only the identity element of G. The principal open question in connection with these groups is whether non-solvable FPF-groups exist. One of the results of the present paper is that if a Sylow p-group of the FPF-group G is the direct product of any number of mutually non-isomorphic cyclic groups, then G has a normal p-complement. As a consequence of this, the conjecture that all FPF-groups are solvable would be true if it were true that every finite simple group has a non-trivial SylowT subgroup of the kind just described. Here it should be noted that all the known simple groups satisfy this property.


1974 ◽  
Vol 17 (1) ◽  
pp. 129-130 ◽  
Author(s):  
Gerald Losey

Let G be a group, ZG its integral group ring and U(ZG) the group of units of ZG. The elements ±g∈U(ZG), g∈G, are called the trivial units of ZG. In this note we will proveLet G be a finite group. If ZG contains a non-trivial unit of finite order then it contains infinitely many non-trivial units of finite order.In [1] S. D. Berman has shown that if G is finite then every unit of finite order in ZG is trivial if and only if G is abelian or G is the direct product of a quaternion group of order 8 and an elementary abelian 2-group.


1999 ◽  
Vol 60 (2) ◽  
pp. 177-189 ◽  
Author(s):  
K.W. Gruenberg ◽  
L.G. Kovács

Let G be a finite group, F a free group of finite rank, R the kernel of a homomorphism φ of F onto G, and let [R, F], [R, R] denote mutual commutator subgroups. Conjugation in F yields a G-module structure on R/[R, R] let dg(R/[R, R]) be the number of elements required to generate this module. Define d(R/[R, F]) similarly. By an earlier result of the first author, for a fixed G, the difference dG(R/[R, R]) − d(R/[R, F]) is independent of the choice of F and φ; here it is called the proficiency gap of G. If this gap is 0, then G is said to be proficient. It has been more usual to consider dF(R), the number of elements required to generate R as normal subgroup of F: the group G has been called efficient if F and φ can be chosen so that dF(R) = dG(R/[R, F]). An efficient group is necessarily proficient; but (though usually expressed in different terms) the converse has been an open question for some time.


2012 ◽  
Vol 55 (1) ◽  
pp. 9-21 ◽  
Author(s):  
COY L. MAY ◽  
JAY ZIMMERMAN

AbstractLet G be a finite group. The symmetric genus σ(G) is the minimum genus of any Riemann surface on which G acts faithfully. We show that if G is a group of order 2m that has symmetric genus congruent to 3 (mod 4), then either G has exponent 2m−3 and a dihedral subgroup of index 4 or else the exponent of G is 2m−2. We then prove that there are at most 52 isomorphism types of these 2-groups; this bound is independent of the size of the 2-group G. A consequence of this bound is that almost all positive integers that are the symmetric genus of a 2-group are congruent to 1 (mod 4).


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