304 research outputs found
Nonsoluble Length Of Finite Groups with Commutators of Small Order
Let p be a prime. Every finite group G has a normal series each of whose
quotients either is p-soluble or is a direct product of nonabelian simple
groups of orders divisible by p. The non-p-soluble length of G is defined as
the minimal number of non-p-soluble quotients in a series of this kind.
We deal with the question whether, for a given prime p and a given proper
group variety V, there is a bound for the non-p-soluble length of finite groups
whose Sylow p-subgroups belong to V. Let the word w be a multilinear
commutator. In the present paper we answer the question in the affirmative in
the case where p is odd and the variety is the one of groups in which the
w-values have orders dividing a fixed number
Generalised quadrangles with a group of automorphisms acting primitively on points and lines
We show that if G is a group of automorphisms of a thick finite generalised
quadrangle Q acting primitively on both the points and lines of Q, then G is
almost simple. Moreover, if G is also flag-transitive then G is of Lie type.Comment: 20 page
On the Exponent of a Verbal Subgroup in a Finite Group
Let w be a multilinear commutator word. We prove that if e is a positive
integer and G is a finite group in which any nilpotent subgroup generated by
w-values has exponent dividing e then the exponent of the corresponding verbal
subgroup w(G) is bounded in terms of e and w only.Comment: Accepted for publication in the Journal of the Australian
Mathematical Societ
Classification of skew translation generalized quadrangles, I
We describe new classification results in the theory of generalized quadrangles (= Tits-buildings of rank 2 and type B-2), more precisely in the (large) subtheory of skew translation generalized quadrangles ("STGQs"). Some of these involve, and solve, long-standing open problems
Bounding the Exponent of a Verbal Subgroup
We deal with the following conjecture. If w is a group word and G is a finite
group in which any nilpotent subgroup generated by w-values has exponent
dividing e, then the exponent of the verbal subgroup w(G) is bounded in terms
of e and w only. We show that this is true in the case where w is either the
nth Engel word or the word [x^n,y_1,y_2,...,y_k] (Theorem A). Further, we show
that for any positive integer e there exists a number k=k(e) such that if w is
a word and G is a finite group in which any nilpotent subgroup generated by
products of k values of the word w has exponent dividing e, then the exponent
of the verbal subgroup w(G) is bounded in terms of e and w only (Theorem B)
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