2-Transitive permutation groups by Mazurov V. D. PDF

By Mazurov V. D.

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The sort of result we have in mind here, in contrast to the type of result to be proved in the earlier part of Chapter 3, is for example: in such-and-such a group the maximal p-subgroups are all conjugate. In order to obtain such a result heavy restrictions have to be imposed upon the group and a large number of nasty examples of every shade and hue have been amassed. See for example K O V ~ CNeumann S, and de Vries [l 1, B. 3 and Heineken’s example given in the Introduction. Because of this the usefulness of such results seems to be virtually nil, at least to date, and this topic forms no natural part of our narrative.

2 Lemma. Every group satisfying rnin is periodic. By considering an infinite, elementary abelian p-group, one realises that Min # LMin (otherwise, of course, the class L s of all locally finite groups would be contained in Mm). But fortunately, a slightly restricted local theorem still holds. 3 Lemma. If every countable subgroup of the group G satisjes min, then G E Min. 30 BASIC METHODS, CONCEPTS AND EXAMPLES [CH. 1, 9E Proof. If this were not so, then in the group G there would be an infinite, strictly descending chain {Gi}, Gi 3 G i + l , of subgroups.

5. 5. 4 can be found in Blackburn [l] and, at least implicitly, in the works of Cernikov. 6 Proposition. If a locally Jinite p-group G contains a maximal elementary abelian subgroup E which is$nite, then G is a Cernikov group. CH. 1, 8 GI THE MINIMAL CONDITION ON ABELIAN SUBGROUPS 41 Proof. Consider for the moment a vector space V over a field of characteristic p and a finite elementary abelian p-group E of automorphisms of V. We claim that C,E has dimension at least [IEI-' dim V ] (trivially it is a subspace).

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