On the groups generated by a class of \(p\)-involutions.
A group of order \(2p^ n\) (\(p\) prime) is said to be an \(F(p)\)-group if it is spanned by its involutions. These groups have been considered by \textit{B. Fischer} [Math. Z. 83, 267-303 (1964; Zbl 0119.26201)], and (for \(p=3)\) by \textit{L. Bénéteau} [C. R. Acad. Sci., Paris, Sér. A 285, 735-737 (1977; Zbl 0389.20026)]. Let \(P\) be a \(p\)-group. It is shown that there is an \(F(p)\)-group \(G\) such that \(D(G)=P\) if and only if \(P\) has an automorphism \(f\) such that \(P=\langle x\mid f(x)=x^{-1}\rangle\). Whenever it exists, this \(G\) is unique up to isomorphism. If \(P\) has an automorphism \(f\) as above, this holds also for \(P\wr C_ q\) (\(q\) a power of \(p\)). Finally, every finite \(p\)-group can be embedded in an \(F(p)\)-group.
scientific article; zbMATH DE number 4122210