On the solution of certain equations with exponent sum 0 over \(\mathbb{Z}_2\)
Let \(G\) be a group and \(t\) an unknown. Then a solution of an equation \(g_0t^{e(1)}g_1\cdots t^{e(n)}g_n=1\) in \(t\) over \(G\) is an embedding of \(G\) into a group \(H\), together with an element \(h\) of \(H\), such that \(g_0h^{e(1)}g_1\cdots h^{e(n)}g_n=1\) in \(H\). \(\sigma=e(1)+\cdots+e(n)\) is called the exponent sum. When \(\sigma=0\), nonsolvable examples are known [see \textit{R. C. Lyndon}, Bol. Soc. Bras. Math. 11, No. 1, 79-102 (1980; Zbl 0463.20030)].NEWLINENEWLINENEWLINEIn the paper under review the solvability of certain equations over \(\mathbb{Z}_2\) with exponent sum zero is examined. The equations are generalizations of the equation studied by Lyndon.
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