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Wreath decompositions of finite permutation groups

Published online by Cambridge University Press:  17 April 2009

L. G. Kovács
Affiliation:
Mathematics IASAustralian National UniversityGPO Box 4Canberra 2601Australia
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Abstract

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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.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1989

References

[1]Gross, Fletcher and Kovács, L. G., ‘On normal subgroups which are direct products’, J. Algebra 90 (1984), 133168.CrossRefGoogle Scholar
[2]James, Gordon and Kerber, Adalbert, The representation theory of the symmetric group (Addison-Wesley, Reading, 1981).Google Scholar
[3]Kovács, L. G., ‘Primitive subgroups of wreath products in product action’, Proc. London Math. Soc. (3) 58 (1989), 306322.CrossRefGoogle Scholar
[4]Liebeck, Martin W., Praeger, Cheryl E. and Saxl, Jan, ‘On the O'Nan-Scott Theorem for finite primitive permutation groups’, J. Austral. Math. Soc. Ser. A 44 (1988), 389396.CrossRefGoogle Scholar
[5]Praeger, Cheryl E., ‘The inclusion problem for finite primitive permutation groups’, Proc. London Math. Soc. (3) 59 (to appear).Google Scholar