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Chief Series and Right Regular Representations of Finite p-Groups

Part of: Representation theory of groups Permutation groups

Published online by Cambridge University Press:  09 April 2009

Felix Leinen
Felix Leinen
Affiliation:
Fachbereich 17-Mathematik Johannes-Gutenberg-Universität Saarstr. 21 6500 Mainz West, Germany
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Abstract

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We study the embeddings of a finite p-group U into Sylow p-subgroups of Sym (U) induced by the right regular representation p: U→ Sym(U). It turns out that there is a one-to-one correspondence between the chief series in U and the Sylow p-subgroups of Sym (U) containing Up. Here, the Sylow p-subgroup Pσ of Sym (U) correspoding to the chief series σ in U is characterized by the property that the intersections of Up with the terms of any chief series in Pσ form σp. Moreover, we see that p: U→ Pσ are precisely the kinds of embeddings used in a previous paper to construct the non-trivial countable algebraically closed locally finite p-groups as direct limits of finite p-groups.

MSC classification

Secondary: 20D15: Nilpotent groups, $p$-groups 20B35: Subgroups of symmetric groups
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 44 , Issue 2 , April 1988 , pp. 225 - 232
Copyright
Copyright © Australian Mathematical Society 1988

References

[1]Hall, P., ‘Some constructions for locally finite groups’, J. London Math. Soc. 34 (1959), 305–319.CrossRefGoogle Scholar
[2]Hall, P., ‘On the embedding of a group in a join of given groups’ J. Austral. Math. Soc. 17 (1974), 434–495.Google Scholar
[3]Leinen, F., ‘A uniform way to control chief series in finite p-groups and to construct the countable algebraically closed locally finite p-groups’, J. London Math. Soc. (2) 33 (1986), 260–270.CrossRefGoogle Scholar
[4]Suprunenko, D., Soluble and nilpotent linear groups (Translations of Mathematical Monographs 9, Providence, R.I., 1963).Google Scholar