Infinite alternating groups as finitary linear transformation groups

Abstract

abstract degree of a classical group as its natural projective representation degree (see (6.2)). Case (1) of (2.6) is dispensed with using Mal’cev’s local characterization of finite dimensional linear groups and their classification. THEOREM (2.7) (Mal’cev, see [ 13, l.L.91). Let G be a Iocally finite group with adeg(G) < 00. Then G is a finite dimensional linear group. THEOREM (2.8) [l, 2,8,21]. Let G be an infinite group. Then is a simple, locally finite, linear group of finite degree if and only if G is isomorphic to an adjoint group of Lie type over an infinite subfield Fp, the algebraic closure of [F,, for some prime p. 3. LOCAL CHARACTERIZATION OF FINITARY LINEAR GROUPS; A THEOREM OF MAL’CEV TYPE Mal’cev’s theorem (2.7) provides a local characterization of locally finite linear groups G of finite degree. The property of having a faithful linear representation of degree n passes from the set of finite subgroups of G to G itself. Under this section we provide a similar result for locally finite simple groups which are linitary linear. Remember that the linear transformation g of the vector space I/ is finitary if g fixes pointwise a subspace of finite codimension in V. That is, the subspace C,(g) = {v E V 1 vg = v} has finite codimension in V, or equivalently the subspace [V, g] = {v(g - 1) 1 v E V} has finite dimension in V. The collection of all linitary members of GL( V) forms a normal sub- group FGL( V), and a group which is (isomorphic to) a subgroup of FGL( V), for some V, is sometimes called a finitary linear group.