The object of this note is to point out and correct an error in the paper [2] of Fox, purporting to prove Fenchel's conjecture that a finitely generated, infinite Fuchsian group has a torsion-frcc normal subgroup of finite index. Fox divided the proof of the main result in his paper into four cases and an error occurs in the proofs of Cases (III) and (IV). A direct proof of Case (III) was given. While a direct proof of Case (IV) can be given, the author shows that it also follows indirectly from the result in the paper [1] of Burns and Solitar.Given any three integers a > 1, b > 1 and c > 1, there exist permutations A of order a and B of order b, such that AB has order c.Fox divided the proof of this lemma into the four cases:(I) ab = 0 (mod 2) and max{a, b) < c < a + b, (II) ab = 0 (mod 2) and c > a + b, (III) a = 6 = c =. 1 (mod 2), (IV) a =. b = 1 and c = 0 (mod 2), and subcases. In the subcase m(a + b -2) + 2 < c ^ (m + l)(b + 1) -1 of Case (III) where m is the largest integer not exceeding (c -l)/(b + 1), and it is assumed that a < b < c, the number of distinct symbols in the permutations [2, p. 64] m B= (v\ ■•■Ü>¿_,_, ■■■w\p2) ■ u («X-2 •••w'lPl+l).
Communicated by G. E. WallWe shall take for granted the basic terminology currently in use in the theory of varieties of groups. Kovacs, Newman, Pentony [2] and Levin [3] prove that if m is an integer greater than 2, then the variety N m of all nilpotent groups of class at most m is generated by its free group f m _ 1 (N IB ) of rank m -1 but not by its free group F m _ 2 (NJ of rank m -2. That is, the free groups F k (N m ), 2^k m -2, do not generate N m . In general little is known of th: varieties generated by them. The purpose of the present paper is to record the varieties of the free groups F*(N m ) of the nilpotent varieties N m of all nilpotent groups of class at most m f o r 2^/ c^m -2 and 5 ^ m ^ 6.
It is known ([3], p. 100) that every nilpotent variety of class m is generated by its free group of rank m . This applies in particular to The purpose of the present thesis is to determine the varieties of some free groups of small rank, namely, ^2(^5) »
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