zbMATH — the first resource for mathematics

On the generation gap of a finite group. (English) Zbl 0594.20004
The generation gap of a finite group G, denoted by gap(G), is defined as the difference between d(G), the minimal number of generators of G, and \(d_ G(I(G))\), the minimal number of generators of the augmentation ideal I(G) as a module over the integral group ring \({\mathbb{Z}}G\). The author obtains certain group-theoretical bounds for gap(G). For a subgroup \(S\subset G\) he defines an integer d(G,S) equal to the minimal number of elements of G needed to generate G together with S. Then for any G there is a prime \(p| | G|\) such that for every subgroup H of order prime to p the inequality gap(G)\(\geq d(G)-d(G,H)-1\) holds. If p and G are such that each non-abelian composition factor of G can be generated by two of its Sylow p-subgroups then for any Sylow p-subgroup \(S\subset G\), d(G)-d(G,S)\(\geq gap(G)\). For many groups these inequalities give an estimation of gap(G) up to \(\pm 1\).
Reviewer: S.Jackowski

20C05 Group rings of finite groups and their modules (group-theoretic aspects)
20F05 Generators, relations, and presentations of groups
20J06 Cohomology of groups
20D60 Arithmetic and combinatorial problems involving abstract finite groups
Full Text: DOI
[1] Aschbacher, M.; Guralnick, R., Solvable generation of groups and Sylow subgroups of the lower central series, J. algebra, 77, 189-201, (1982) · Zbl 0485.20012
[2] Aschbacher, M.; Guralnick, R., Some applications of the first cohomology group, J. algebra, 90, 446-460, (1984) · Zbl 0554.20017
[3] Auslander, M., Relative cohomology theory of groups and continuations of homomorphism, ()
[4] Baumann, B., Endliche gruppen, die von je zwei verschiedenen ihrer 2-sylowuntergruppen erzeugt werden, Arch. math., 28, 34-40, (1977) · Zbl 0387.20012
[5] Cossey, J.; Gruenberg, K.W.; Kovacs, L., The presentation rank of a direct product of finite groups, J. algebra, 28, 597-603, (1974) · Zbl 0293.20033
[6] Gaschütz, W., Über modulare darstellungen endlicher gruppen, die von freien gruppen induziert werden, Math. Z., 60, 274-286, (1954) · Zbl 0056.02401
[7] Gruenberg, K.W., Relation modules of finite groups, () · Zbl 0513.20039
[8] Gruenberg, K.W., Groups of non-zero presentation rank, (), 215-224 · Zbl 0358.20045
[9] Gruenberg, K.W., Free abelianised extensions of finite groups, (), 71-103 · Zbl 0513.20039
[10] Hilton, P.J.; Stammbach, U., A course in homological algebra, () · Zbl 0238.18006
[11] Jacobinski, H., Representation of orders over a Dedekind domain, Symposia math., 15, 397-401, (1975) · Zbl 0324.20008
[12] Kimmerle, W., Relative relation modules as generators for integral group rings of finite groups, Math. Z., 1972, 143-156, (1980) · Zbl 0415.20002
[13] Kimmerle, W., Growth sequences relative to subgroups, (), 252-260 · Zbl 0524.20004
[14] Kimmerle, W.; Williams, J.S., On minimal relation modules and 1-cohomology of finite groups, Arch. math., 42, 214-223, (1984) · Zbl 0553.20003
[15] Kovacs, L., On finite soluble groups, Math. Z., 103, 37-39, (1968) · Zbl 0183.02804
[16] Linnell, P.A., Relation modules and augmentation ideals of finite groups, J. pure appl. algebra, 22, 143-164, (1981) · Zbl 0476.20008
[17] Roggenkamp, K.W., Relation modules of finite groups and related topics, Alg. i logika, 12, 351-359, (1973) · Zbl 0286.20045
[18] Roggenkamp, K.W., Integral representations and presentations of finite groups, (), 2nd part · Zbl 0955.16020
[19] Swan, R.G., Minimal resolutions for finite groups, Topology, 4, 193-208, (1965) · Zbl 0146.04002
[20] Takasu, S., Relative homology theory and relative cohomology theory of groups, J. fac. sci. univ. Tokyo, sect. I, 8, 75-110, (1959) · Zbl 0092.02203
[21] Wiegold, J., Growth sequences of finite groups II, J. austr. math. soc., 17, 133-141, (1974) · Zbl 0286.20025
[22] Wielandt, H., Finite permutation groups, (1964), Academic Press New York · Zbl 0138.02501
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.