On the Abstract Groups (3, N, P; 2)

The groups (m;n; p; q) = ha; b j a = b = (ab) = [a; b] = 1i were introduced and studied by Brahana [?, ?], Coxeter [?, ?, ?, ?] and Sinkov [?, ?, ?, ?, ?]. This initial work was followed by a later paper of Sinkov [?], and a paper by Leech and Mennicke [?]. More recently, interest in deciding which values of the parameters m;n; p; q yield nite groups and which in nite has been revived by Grove and McShane [?], Chaltin [?], Howie and Thomas [?], Holt and Plesken [?], and Edjvet [?, ?]. Since any permutation of m;n; p does not change the resulting group, we may assume that 2 m n p. The case m = n = 2 is straightforward, and the case (m;n) = (2; 3) has been investigated by Howie and Thomas [?], Holt and Plesken [?], and Edjvet [?]. The only group in this category which remains unresolved is (2; 3; 13; 4), which is known to have a homomorphic image E PSL(2; 25) of order 219:34:52:132, where E is an extension of the elementary abelian group of order 212 by PSL(3; 3), but it has not yet been shown to be in nite. (See [?] for further discussion of this group.) With the above exception, Edjvet [?] has completely determined which groups (m;n; p; q) are nite and which in nite, provided that (m; q) 6= (3; 2). This leaves the groups (3; n; p; 2), and these