A note on Itô’s theorem of \(p\)-nilpotence. (English) Zbl 1490.20016
A finite group \(G\) is \(p\)-nilpotent if there exists a normal subgroup \(H\) of order coprime to \(p\) such that \([G:H]\) is a \(p\)-power. If \(P\) is a \(p\)-group, then define
\begin{align*}
\Omega(P) &= \begin{cases} \langle x\in G ~|~ x^p=1\rangle &\text{if \(p>2\) or \(P\) abelian}, \\
\langle x\in G ~|~ x^4=1\rangle &\text{otherwise}. \end{cases}
\end{align*}
A result of N. Ito [Kōdai Math. Semin. Rep. 1951, 1–6 (1951; Zbl 0044.01303)] states that a group \(G\) is \(p\)-nilpotent if \(\Omega(P)\) is central in \(G\), where \(P\) is a Sylow \(p\)-group of \(G\). T. J. Laffey showed in [Proc. Camb. Philos. Soc. 75, 133–137 (1974; Zbl 0277.20022)] that \(G\) is \(p\)-nilpotent if and only if \(N_G(P)\) is \(p\)-nilpotent, assuming that \(\Omega(P)\) is central in \(P\).
The authors generalize the setting and ask about the \(p\)-nilpotency of \(G\) provided \(\Omega(P)\) is contained in the norm of \(P\). The norm of a group \(G\), denoted by \(N(G)\), is the intersection of all \(N_G(H)\) for every \(H\leqslant G\). They show that if \(p\) is an odd prime, \(P\) a Sylow \(p\)-subgroup of \(G\), and \(\Omega(P)\leqslant N(P)\), then \(G\) is \(p\)-nilpotent if and only if \(N_G(P)\) is \(p\)-nilpotent. The authors prove additional results in this direction under various generalizations.
The authors generalize the setting and ask about the \(p\)-nilpotency of \(G\) provided \(\Omega(P)\) is contained in the norm of \(P\). The norm of a group \(G\), denoted by \(N(G)\), is the intersection of all \(N_G(H)\) for every \(H\leqslant G\). They show that if \(p\) is an odd prime, \(P\) a Sylow \(p\)-subgroup of \(G\), and \(\Omega(P)\leqslant N(P)\), then \(G\) is \(p\)-nilpotent if and only if \(N_G(P)\) is \(p\)-nilpotent. The authors prove additional results in this direction under various generalizations.
Reviewer: Joshua Maglione (Magdeburg)
MSC:
| 20D15 | Finite nilpotent groups, \(p\)-groups |
References:
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