caa a22 4500 386360154 CHVBK 20180307111912.0 cr unu---uuuuu 161130e198910 xx s 000 0 eng 10.1017/S0004972700004366 doi S0004972700004366 pii (NATIONALLICENCE)cambridge-10.1017/S0004972700004366 Kovács L. G. Mathematics IAS Australian National University GPO Box 4 Canberra 2601 Australia Wreath decompositions of finite permutation groups [Elektronische Daten] [L. G. Kovács] There is a familiar construction with two finite, transitive permutation groups as input and a finite, transitive permutation group, called their wreath product, as output. The corresponding ‘imprimitive wreath decomposition' concept is the first subject of this paper. A formal definition is adopted and an overview obtained for all such decompositions of any given finite, transitive group. The result may be heuristically expressed as follows, exploiting the associative nature of the construction. Each finite transitive permutation group may be written, essentially uniquely, as the wreath product of a sequence of wreath-indecomposable groups, amid the two-factor wreath decompositions of the group are precisely those which one obtains by bracketing this many-factor decomposition. If both input groups are nontrivial, the output above is always imprimitive. A similar construction gives a primitive output, called the wreath product in product action, provided the first input group is primitive and not regular. The second subject of the paper is the ‘product action wreath decomposition' concept dual to this. An analogue of the result stated above is established for primitive groups with nonabelian socle. Given a primitive subgroup G with non-regular socle in some symmetric group S, how many subgroups W of S which contain G and have the same socle, are wreath products in product action? The third part of the paper outlines an algorithm which reduces this count to questions about permutation groups whose degrees are very much smaller than that of G. Copyright © Australian Mathematical Society 1989 Bulletin of the Australian Mathematical Society Cambridge University Press 40/2(1989-10), 255-279 0004-9727 40:2<255 1989 40 BAZ https://doi.org/10.1017/S0004972700004366 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1017/S0004972700004366 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Kovács L. G. Mathematics IAS Australian National University GPO Box 4 Canberra 2601 Australia NATIONALLICENCE 773 0- Bulletin of the Australian Mathematical Society Cambridge University Press 40/2(1989-10), 255-279 0004-9727 40:2<255 1989 40 BAZ CC0 http://creativecommons.org/publicdomain/zero/1.0 nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-cambridge