zbMATH — the first resource for mathematics

On certain products of soluble groups. (English) Zbl 0984.20016
The authors consider groups \(G=HK\) which are the product of two soluble subgroups \(H\) and \(K\) and satisfy additional permutability requirements, namely that certain subgroups of \(H\) commute with certain subgroups of \(K\). Under these conditions the structure of the factorized group \(G\) is investigated, and it is shown that certain group theoretical properties (in particular finiteness conditions) are inherited by these products. Also bounds for the derived length of these products are given in some cases.

20E15 Chains and lattices of subgroups, subnormal subgroups
20F16 Solvable groups, supersolvable groups
20D40 Products of subgroups of abstract finite groups
20E22 Extensions, wreath products, and other compositions of groups
20F14 Derived series, central series, and generalizations for groups
Full Text: DOI
[1] Amberg, B., Franciosi, S. and de Giovanni, F.: Products of Groups. Clarendon Press, Oxford 1992 · Zbl 0774.20001
[2] Assad M., Arch. Math. 53 pp 318– (1989)
[3] Beidleman J., Journal of Group Theory 2 pp 377– (1999)
[4] Carocca A., Arch. Math. 71 pp 257– (1998)
[5] Carocca, A. and Maier, R.: Theorems of Kegel-Wielandt type. Groups St. Andrews 1997 in Bath I, 195-201. Cambridge University Press · Zbl 0929.20020
[6] Cohn P., Arch. Math. 7 pp 94– (1956)
[7] Heineken H., Arch. Math. 41 pp 498– (1983)
[8] Howlett, R. B.: On the exponent of certain factorizable groups. J. London Math. Soc. (2) 31 (1985), 265-271 · Zbl 0526.20015
[9] Huppert B., Nagoya Math. J. 6 pp 93– (1953)
[10] Ito N., Math. Z. 62 pp 400– (1955)
[11] Jabara, E.: Una nota sulla struttura di alcuni gruppi fattoriali. Bolletino U. M. I. XI-B (1997), 351-354
[12] Kegel O. H., Math. Z. 87 pp 42– (1965)
[13] Lennox, J. C.: On the solubility of a poduct of permutable subgroups. J. Austral. Math. Soc. 22 (Series A) (1976), 252-255 · Zbl 0342.20016
[14] Lennox, J. C. and Stonehewer, S. E.: Subnorml Subgroups of Groups. Oxford University Press 1987 · Zbl 0606.20001
[15] Sesekin N. F., Sib. Mat. J. 9 pp 1070– (1968)
[16] Sesekin N. F., Math. Notes 13 pp 266– (1973)
[17] Stonehewer S. E., Math. Z. 125 pp 1– (1972)
[18] Stonehewer S. E., J. Austral. Math. Soc. 16 pp 90– (1973)
[19] Wehrfritz, B. A. F.: In nite Linear Groups. Springer, Berlin 1973
[20] Zaitsev D. I., Algebra and Logic 19 pp 94– (1980)
[21] Zaitsev D. I., Akad. Nauk Ukrain. Inst. Mat. Kie pp 13– (1982)
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.