INTRODUCTION

It has been known for some time [1, Ch.18] that finitely generated groups of exponent 2, 3, 4 or 6 are finite. So far there is much less known on groups of exponent 5. It was shown by Kostrikin in 1955 that the largest finite group with two generators has order at most 534. In 1956 Graham Higman used a combinatorial argument to show that for any finite number of generators there is a largest finite group of exponent 5. In 1974 Havas, Wall and Warns ley showed that the largest finite two generator group of exponent 5 has order exactly 534 and they found a detailed table of commutator relations which describe this group exactly.

In this paper in an attempt to prove finite B(5,2) the Burnside group of exponent 5 with two generators, it is shown that B(5,2) has a normal subgroup K1 of index 510. K1 is the normal closure of 24 explicit elements. K1 should be an Abelian group of order 524. This reduces the proof of the finiteness of B(5,2) to a proof that K1 is Abelian.