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Journal of the London Mathematical Society 1997 55(2):335-340; doi:10.1112/S0024610796004796
© 1997 by London Mathematical Society
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© The London Mathematical Society

Linear Groups Generated by Elements of Small Degree

Richard E. Phillips

Department of Mathematics, Michigan State University East Lansing, Michigan 48824, USA. E-mail: rphillips{at}math.msu.edu

Received 12 April 1996.

If F is a subset of G{subseteq}GL (n, K)=GL(V, K) (where K is a field) the degree of F(=deg (F)) is the dimension of the K-space [V, F] spanned by

{v(f–1)|visinV, fisin<F>};

note that in the special case F={g} we have [V, g]={v(g–1)|visinV}. Our intention is to describe those irreducible linear groups G{subseteq}GL (n, K) generated by elements whose degrees are small relative to n. To do this successfully it seems necessary to work within the restricted class of ‘solvable-by-locally finite’ groups (throughout, this class will be denoted by S(LF)). Somewhat surprisingly, it turns out that if G is an irreducible S(LF)-subgroup of GL (n, K) generated by elements of small degree (relative to n), then G has large non-abelian simple sections. For a linear group G, the restriction to the class S(LF) is equivalent to insisting that G have no non-cyclic free subgroups (see [7, Section 5.6]). Our main result in this direction is the following structure theorem.


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