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

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.

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

scholarly journals Bounding the maximal size of independent generating sets of finite groups

2020 ◽  
pp. 1-18
Author(s):  
Andrea Lucchini ◽  
Mariapia Moscatiello ◽  
Pablo Spiga
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Prime Numbers ◽  
Generating Set ◽  
Generating Sets ◽  
Number Of Generators ◽  
Maximal Size ◽  
A Finite Group ◽  
Minimal Generating Set

Abstract Denote by m(G) the largest size of a minimal generating set of a finite group G. We estimate m(G) in terms of $\sum _{p\in \pi (G)}d_p(G),$ where we are denoting by d p (G) the minimal number of generators of a Sylow p-subgroup of G and by π(G) the set of prime numbers dividing the order of G.

On the Minimum Order of Graphs with Given Group

1974 ◽  
Vol 17 (4) ◽  
pp. 467-470 ◽  
Author(s):  
László Babai
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Upper Bound ◽  
Minimum Order ◽  
Number Of Generators ◽  
Minimum Number ◽  
A Finite Group

For G a, finite group let α(G) denote the minimum number of vertices of the graphs X the automorphism group A(X) of which is isomorphic to G.G. Sabidussi proved [1], that α(G)=0(n log d) where n=\G\ and d is the minimum number of generators of G.As 0(log n) is the best possible upper bound for d, the result established in [1] implies that α(G)=0(n log log n).

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 HIGHER FROBENIUS–SCHUR INDICATORS

2021 ◽  
pp. 1-11
Author(s):  
YANJUN LIU ◽  
WOLFGANG WILLEMS
Keyword(s):  
Finite Group ◽  
Representation Theory ◽  
Irreducible Characters ◽  
A Finite Group

Abstract Similarly to the Frobenius–Schur indicator of irreducible characters, we consider higher Frobenius–Schur indicators $\nu _{p^n}(\chi ) = |G|^{-1} \sum _{g \in G} \chi (g^{p^n})$ for primes p and $n \in \mathbb {N}$ , where G is a finite group and $\chi $ is a generalised character of G. These invariants give answers to interesting questions in representation theory. In particular, we give several characterisations of groups via higher Frobenius–Schur indicators.

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