LINEAR GROUPS GENERATED BY ELEMENTS OF SMALL DEGREE

Abstract

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

{v(f−1)[mid ]v∈V, f∈〈F〉};

note that in the special case F={g} we have [V, g]={v(g−1)[mid ]v∈V}. Our intention is to describe those irreducible linear groups G⊆GL (n, [Kscr ]) 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 [Sscr ](L[Fscr ])). Somewhat surprisingly, it turns out that if G is an irreducible [Sscr ](L[Fscr ])-subgroup of GL (n, [Kscr ]) 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 [Sscr ](L[Fscr ]) 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.