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Using the fundamental group of a punctured torus, a free group F of rank two, and the fact that the natural eipmorphism from AutF onto Aut(F/F') has as kernel the group of inner automorphisms of F, we describe representatives of the conjugacy classes of generating pairs of F and give explicit relations between them. Let F = F(S, T) be the free group on S and T. By a theorem of Nielsen [N] (see [LS, p. 25]) the natural epimorphism from AutF onto Aut(F/F') ( = GL(2, Z)) has as kernel the group of inner automorphisms of F. From this it follows easily that, if a is the abelianization homomorphism from F onto F/F' (= Z2) and a E 22 is primitivel, then the inverse image of a under a is a conjugacy class of primitive elements. Also, if (ai,a2) is a basis of 22, then, up to conjugacy, there is a unique basis (fi,f2) of F such that (fi)a = ai (i = 1,2). (The basis (fi, f2) is conjugate to (gl, 2) if there exists w E F such that w-lfiw = gi (i = 1,2)). In the important paper [OZ], Osborne and Zieschang define explicitly primitive words Wm,n E F(S,T), where m and n are relatively prime integers, such that (Wm,n)a = (m, n). They also state that if mn pq = 1, then (Wm,n, Wp,q) is a basis of F; this, while correct for nonnegative values of m,n,p, q, is not valid in general (for example W-2,-3 and W1,1 do not generate F). A composition formula is also stated in [OZ, Thm. 3.5] but this, even with the correction of indices in [LTZ, 2.1.3], is incorrect in general. In the present article we consider elements Va of F for a = (m, n) E Z2 and e E D C IR2 where D is the complement of the union of all the lines that intersect 22 in more than one point. If gcd(m,n) = 1, then V(,) is conjugate to Wm,n. We show in Theorem 1.i) that (Va, Vb) is a basis of F, if Z2 = (a, b), and obtain in Theorem l.ii) a composition formula. Everything is obtained by applying the fundamental group functor ir to the punctured torus. Denote by T the torus IR2/ 22, by To the punctured torus (R2 _ Z2) / Z2 and by p : R2 Z2 -To the natural projection. If a E 22 and e E D, then denote (e)p by E and define ya E 7r(To, ) as the homotopy class of the loop (e+ta)p, t E [0,1]. Denote ' o0) (resp. y0 ,1)) by S (resp. Te). There is an isomorphism Received by the editors September 15, 1997. 1991 Mathematics Subject Classification. Primary 57M07, 20E05.

A composition formula in the rank two free group