Free Abelian X-groups (Q1075435)

Let \(X\) be an associative ring. An \(X\)-group is a group \(G\) equipped with an action \(G\times X\to G\), \((g,x)\to g^ x\) such that \(g^ 1=g\), \((g^ x)^ y=g^{xy}\), \(g^ xg^ y=g^{x+y}\) for any \(x,y\in X\) and \(g\in G\). These groups were introduced by \textit{R. C. Lyndon} [Trans. Am. Math. Soc. 96, 518-533 (1960; Zbl 0108.02501)]. Let \(X\) be an integral polynomial ring. The author gives a new proof of the following result of Lyndon. Theorem: In a free abelian \(X\)-group the word problem is solvable. The second theorem shows the existence of continuously many nonisomorphic 2-generated abelian \(X\)-groups.