Representations of Locally Finite Groups

The purpose of this paper is to give a brief general account of the completely reducible finite-dimensional representations of a locally finite group G over a given algebraically closed field K. Theorem 1 shows that all such representations of G can be brought down to the algebraic closure F in K of the prime field of K. This reduces all further considerations in this account to countable groups. Theorem 2 characterizes the existence of a faithful completely reducible representation of G of degree n over K in terms of the existence of such representations for appropriate finite subgroups of G. Throughout the paper, G denotes a locally finite group, K denotes an arbitrary algebraically closed field and F denotes the algebraic closure in K of the prime field of K. F denotes an w-dimensional vector space over K. An F-form of V is an -F-subspace W of V such that W and K are linearly disjoint over K and Vis the X-span of W. (Equivalently, an F-form of V is the i^-span of a basis of V.) Il A is an Falgebra, AK denotes the algebra A®pK.