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Let G be a group with generators a" v = l, ■ • • , n. An application of any automorphism A of the free group on the a, or, equivalently, of a sequence of P-transformations (defined below) maps G upon an isomorphic group G'. If G is defined by a set of prescribed relations for the a" G' can be defined by transcribing the original relations in terms of the A_1a,. Even if G is defined by a single relation, it is not known how far the set of all groups with a single defining relation and isomorphic to a given one is determined by the transformations A. However, Grushko's theorem [2]2 implies that at least the decomposibility of G into a free product of two of its proper subgroups can be made obvious by applying a properly chosen A. We shall show that for a G with a single defining relation a result of J. H. C. Whitehead [l] provides a constructive method for finding A and some simple tests for the free indecomposability of G.

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