×

On the structure of \(n\)-isoclinism classes of groups. (English) Zbl 0593.20040

Generalizing an idea of P. Hall, two groups \(G, H\) are called \(n\)-isoclinic if there is an isomorphism \(\alpha\) of \(G/\zeta_ n(G)\) onto \(H/\zeta_ n(H)\) which induces an isomorphism \(\beta\) of \(\gamma_{n+1}(G)\) onto \(\gamma_{n+1}(H)\); here \(\{\gamma_ n(G)\}\), \(\{\zeta_ n(G)\}\) are the lower and upper central series of \(G\) respectively, and \(\beta\) is defined by \([g_ 1,\ldots,g_{n+1}]\beta =[h_ 1,\ldots,h_{n+1}]\) with \(h_ i\zeta_ n(H)=(g_ i\zeta_ n(G))\alpha\). Hall introduced this in the case \(n=1\) and used it to classify various kinds of \(p\)-groups, most notably the groups with an abelian subgroup of index \(p\).
The present paper is concerned with general properties of \(n\)-isoclinism. It was proved in 1963 by Weichsel that two isoclinic groups are always factor groups of a group isoclinic to both, and this was generalized to \(n\)-isoclinism by Bioch. Here it is proved that isoclinic groups can be embedded in a group isoclinic to both. If \(G_ 1\), \(G_ 2\) are \(n\)-isoclinic, \(G_ 1/\zeta_ i(G_ 1)\) and \(G_ 2/\zeta_ i(G_ 2)\) are \((n-i)\)-isoclinic and \(\gamma_{i+1}(G_ 1)\), \(\gamma_{i+1}(G_ 2)\) are \(m\)-isoclinic for an appropriate \(m\). A group \(S\) is called an \(n\)-stem group if \(\zeta_ 1(S)\leq \gamma_{n+1}(S)\); any finitely generated group \(G\) with \(\gamma_{n+1}(G)\) finite is \(n\)-isoclinic to a finite \(n\)-stem group. If \(N\) is a normal subgroup of \(G\) and \(N\cap \gamma_{n+1}(G)=1\), then \(G/N\) is \(n\)-isoclinic to \(G\). A group \(G\) is called quotient irreducible if \(N\cap \gamma_{n+1}(G)=1\) implies \(N=1\); this is equivalent to the condition that the socle of \(G\) is contained in \(\gamma_{n+1}(G)\) and \(\zeta_ 1(G)/(\zeta_ 1(G)\cap \gamma_{n+1}(G))\) is periodic. A similar theorem is proved for subgroups.

MSC:

20E36 Automorphisms of infinite groups
20F14 Derived series, central series, and generalizations for groups
20E34 General structure theorems for groups
20E07 Subgroup theorems; subgroup growth
20D15 Finite nilpotent groups, \(p\)-groups
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Beyl, F. R.; Tappe, J., Group Extensions, Representations and the Schur Multiplier, (Lecture Notes in Math., 958 (1982), Springer: Springer Berlin) · Zbl 0544.20001
[2] Bioch, J. C., Monomiality of groups, Thesis, Leiden (1975) · Zbl 0353.20015
[3] Bioch, J. C., On \(n\)-isoclinic groups, Indag. Math., 38, 400-407 (1976) · Zbl 0363.20011
[4] Bioch, J. C.; van der Waall, R. W., Monomiality and isoclinism of groups, J. reine ang. Math., 298, 74-88 (1978) · Zbl 0366.20004
[5] Hall, M.; Senior, J. K., The Groups of Order \(2^n (n\)≤6) (1964), MacMillan: MacMillan New York
[6] Hall, P., The classification of prime-power groups, J. reine ang. Math., 182, 130-141 (1940) · JFM 66.0081.01
[7] Hall, P., Verbal and marginal subgroups, J. reine ang. Math., 182, 156-157 (1940) · Zbl 0023.29902
[8] Huppert, B., Endliche Gruppen I (1979), Springer: Springer Berlin, Nachdruck der ersten Auflage · Zbl 0412.20002
[9] Huppert, B.; Blackburn, N., Finite Groups II (1982), Springer: Springer Berlin · Zbl 0477.20001
[10] James, R., The groups of order \(p^6 (p\) an odd prime), Math. Comp., 34, 613-637 (1980) · Zbl 0428.20013
[11] Kargapolov, M. I.; Merzljakov, Ju. I., Fundamentals of Groups, (Graduate Texts in Math., 62 (1979), Springer: Springer Berlin) · Zbl 0549.20001
[12] King, S. C., Quotient and subgroup reduction for isoclinism of groups, (Dissertation (1978), Yale University: Yale University New Haven)
[13] MacDonald, I. D., On central series, (Proc. Edinburgh Math. Soc., 13 (1962)), 175-178 · Zbl 0112.25903
[14] Robinson, D. J.S., A Course in the Theory of Groups, (Graduate Texts in Math., 80 (1982), Springer: Springer Berlin) · Zbl 0496.20038
[15] van der Waall, R. W., On the embedding of minimal non-\(M\)-groups, Indag. Math., 36, 157-167 (1974) · Zbl 0279.20013
[16] Weichsel, P. M., On isoclinism, J. London Math. Soc., 38, 63-65 (1963) · Zbl 0114.02103
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.