September 2, 2017

Download 2-Groups which contain exactly three involutions by Konvisser M.W. PDF

By Konvisser M.W.

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Example text

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). For if x E E\

Section H. The minimal condition on normal subgroups Since there are infinite simple groups, the minimal condition on the set of normal subgroups of the group G will not have enough impact to force the group G to be a Cernikov group. One might reasonably cherish such a hope only if G has many normal subgroups. In this section we shall show that this hope is justified for locally nilpotent groups, while even for soluble groups this is not so in general. l. Lemma. Let M be a locally soluble minimal normal subgroup of the group G .

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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