Revisiting the Leinster groups. (Quelques résultats sur les groupes de Leinster.) (English. French summary) Zbl 1292.20026
Given a finite group \(G\), let \(\tau(G)\) be the number of normal subgroups of \(G\) and \(\sigma(G)\) be the sum of the orders of the normal subgroups of \(G\). A finite group \(G\) is a Leinster group if \(\sigma(G)=2\cdot|G|\). Clearly a finite cyclic group \(C_n\) is Leinster if and only if \(n\) is a perfect number, moreover the dihedral Leinster groups are in one-to-one correspondence with odd perfect numbers (one Leinster group of odd order is known: \((C_{127}\rtimes C_7)\times C_n\), \(n=3^4\cdot 11^2\cdot 19^2\cdot 113\)).
In this paper the author classifies Leinster groups \(G\) with \(\tau(G)\leq 7\).
In this paper the author classifies Leinster groups \(G\) with \(\tau(G)\leq 7\).
Reviewer: Enrico Jabara (Venezia)
MSC:
| 20D60 | Arithmetic and combinatorial problems involving abstract finite groups |
| 11A25 | Arithmetic functions; related numbers; inversion formulas |
Keywords:
Leinster groups; perfect numbers; finite groups; orders of normal subgroups; numbers of subgroupsSoftware:
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