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It is proved that a homomorph of the group of trefoil knot cannot be the group of a 2-knot in 4-sphere. In this short note we will give a negative answer to a problem proposed by Fox [2, Problem 28]. Fox asked if a finitely presented group that is a homomorph of a knot group and is infinite cyclic over its commutator subgroup is the group of a locally flat 2-sphere in R4. We prove THEOREM. G = (a, b: a2b-3, [a, b]2) cannot be the group of a locally flat nsphere in (n + 2)-sphere S +2 for n > 2, where [a, b] = aba'lb'-. Since G/G' is infinite cyclic and G is a factor group of G* = (a, b: a2b-3) that is the group of the trefoil knot, our theorem gives a negative answer to the problem, where G' denotes the commutator subgroup of G. Our proof is based on the following PROPOSITION 1 [4, p. 106]. If a finitely presented group X is the group of a locally flat n-sphere Kn in Sn+2 n > 1, i.e. X _ rz (Sn?2 K ), then (1) l/7r' is infinite cyclic, (2) the weight of '7 is one, and (3) H2 (7T) = 0, where H2 (7T) denotes the second homology group of v with integral coefficient and trivial action of 7 on the coefficient group. Furthermore, if n > 3, then (1), (2) and (3) are sufficient conditions for 7T to be the group of some n-knot Kn in Sn+2. Our group G satisfies (1) and (2), since G* does as the knot group. Therefore, to prove the theorem we must show PROPOSITION 2. H2(G) # 0. PROOF. Let F be the free group with two free generators a, b. For a set, X, of elements of F, we denote by XF the normal closure of X in F. Now, let R = {a b ,[a,b 24 . Then GF/R and, by Hopf [3], H2(G) = R n F'/[R, F]. Since [a, b] E F', it suffices to show that [a, b]2 # 1 in the group Received by the editors September 16, 1975. AMS (MOS) subject classifications (1970). Primary 55A25.

On a group that cannot be the group of a 2-knot