scholarly journals The efficiency of PSL(2, p)3 and other direct products of groups

1997 ◽  
Vol 39 (3) ◽  
pp. 259-268 ◽  
Author(s):  
C. M. Campbell ◽  
I. Miyamoto ◽  
E. F. Robertson ◽  
P. D. Williams
Keyword(s):  
Finite Group ◽  
Direct Products ◽  
Schur Multiplier ◽  
Converse Problem ◽  
Number Of Generators ◽  
Products Of Groups ◽  
A Finite Group ◽  
Finite Presentations ◽  
Minimum Deficiency

A finite group G is efficient if it has a presentation on n generators and n + m relations, where m is the minimal number of generators of the Schur multiplier M (G)of G. The deficiency of a presentation of G is r–n, where r is the number of relations and n the number of generators. The deficiency of G, def G, is the minimum deficiency over all finite presentations of G. Thus a group is efficient if def G = m. Both the problem of efficiency and the converse problem of inefficiency have received considerable attention recently; see for example [1], [3], [14] and [15].

On the number of generators of a finite group

1989 ◽  
Vol 53 (6) ◽  
pp. 521-523 ◽  
Author(s):  
Robert M. Guralnick
Keyword(s):  
Finite Group ◽  
Number Of Generators ◽  
A Finite Group

scholarly journals Projective characters of degree one and the inflation-restriction sequence

1989 ◽  
Vol 46 (2) ◽  
pp. 272-280 ◽  
Author(s):  
R. J. Higgs
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Exact Sequence ◽  
Schur Multiplier ◽  
Original Result ◽  
Image Position ◽  
A Finite Group ◽  
Invariant Element

AbstractLet G be a finite group, α be a fixed cocycle of G and Proj (G, α) denote the set of irreducible projective characters of G lying over the cocycle α.Suppose N is a normal subgroup of G. Then the author shows that there exists a G- invariant element of Proj(N, αN) of degree 1 if and only if [α] is an element of the image of the inflation homomorphism from M(G/N) into M(G), where M(G) denotes the Schur multiplier of G. However in many situations one can produce such G-invariant characters where it is not intrinsically obvious that the cocycle could be inflated. Because of this the author obtains a restatement of his original result using the Lyndon-Hochschild-Serre exact sequence of cohomology. This restatement not only resolves the apparent anomalies, but also yields as a corollary the well-known fact that the inflation-restriction sequence is exact when N is perfect.

scholarly journals Proficient presentations and direct products of finite groups

1999 ◽  
Vol 60 (2) ◽  
pp. 177-189 ◽  
Author(s):  
K.W. Gruenberg ◽  
L.G. Kovács
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Free Group ◽  
Finite Groups ◽  
Finite Rank ◽  
Module Structure ◽  
Direct Products ◽  
The Difference ◽  
A Finite Group ◽  
Open Question

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.

A Comment on Finite Nilpotent Groups of Deficiency Zero

1980 ◽  
Vol 23 (3) ◽  
pp. 313-316 ◽  
Author(s):  
Edmund F. Robertson
Keyword(s):  
Finite Group ◽  
Finite Number ◽  
Nilpotent Groups ◽  
Metacyclic Groups ◽  
Generators And Relations ◽  
Number Of Generators ◽  
Nilpotent Class ◽  
A Finite Group

A finite group is said to have deficiency zero if it can be presented with an equal number of generators and relations. Finite metacyclic groups of deficiency zero have been classified, see [1] or [6]. Finite non-metacyclic groups of deficiency zero, which we denote by FD0-groups, are relatively scarce. In [3] I. D. Macdonald introduced a class of nilpotent FD0-groups all having nilpotent class≤8. The largest nilpotent class known for a Macdonald group is 7 [4]. Only a finite number of nilpotent FD0-groups, other than the Macdonald groups, seem to be known [5], [7]. In this note we exhibit a class of FD0-groups which contains nilpotent groups of arbitrarily large nilpotent class.

scholarly journals A NOTE ON p-CENTRAL GROUPS

2013 ◽  
Vol 55 (2) ◽  
pp. 449-456
Author(s):  
RACHEL CAMINA ◽  
ANITHA THILLAISUNDARAM
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Prime Number ◽  
Nilpotency Class ◽  
Schur Multiplier ◽  
Tate Cohomology ◽  
Explicit Bound ◽  
Covering Group ◽  
A Finite Group

AbstractA group G is n-central if Gn ≤ Z(G), that is the subgroup of G generated by n-powers of G lies in the centre of G. We investigate pk-central groups for p a prime number. For G a finite group of exponent pk, the covering group of G is pk-central. Using this we show that the exponent of the Schur multiplier of G is bounded by $p^{\lceil \frac{c}{p-1} \rceil}$, where c is the nilpotency class of G. Next we give an explicit bound for the order of a finite pk-central p-group of coclass r. Lastly, we establish that for G, a finite p-central p-group, and N, a proper non-maximal normal subgroup of G, the Tate cohomology Hn(G/N, Z(N)) is non-trivial for all n. This final statement answers a question of Schmid concerning groups with non-trivial Tate cohomology.

On the number of generators of a finite group

1988 ◽  
Vol 50 (2) ◽  
pp. 110-112 ◽  
Author(s):  
Patrizia Longobardi ◽  
Mercede Maj
Keyword(s):  
Finite Group ◽  
Number Of Generators ◽  
A Finite Group

Classes d'éléments conjugués des groupes cristallographiques

1978 ◽  
Vol 34 (6) ◽  
pp. 895-900
Author(s):  
J. Sivardière
Keyword(s):  
Finite Group ◽  
Double Point ◽  
Invariant Subgroup ◽  
Factor Group ◽  
Direct Products ◽  
Simple Point ◽  
Space Groups ◽  
Point Groups ◽  
A Finite Group

Let G be a finite group, H an invariant subgroup and F the corresponding factor group. The classes of conjugated elements of G are derived from the classes of H and F. We consider simple point groups and symmorphic space groups, which are semi-direct products H^F, then double point groups and non- symmorphic space groups, which are extensions of F by H.

Real projective representations of finite groups

1990 ◽  
Vol 107 (1) ◽  
pp. 27-32 ◽  
Author(s):  
P. N. Hoffmann ◽  
J. F. Humphreys
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Schur Multiplier ◽  
Representations Of Finite Groups ◽  
Projective Representations ◽  
A Finite Group

The projective representations of a finite group G over a field K are divided into sets, each parametrized by an element of the group H2(G, Kx). The latter is the Schur multiplier M(G) when K = ℂ.

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