Criteria for nilpotency of groups via partitions. (English) Zbl 1406.20024
Summary: Let \(G\) be a group and let \(T <G\). A set \(\Pi = \{H_1,H_2, \dots, H_n\}\) of proper subgroups of \(G\) is said to be strict \(T\)-partition of \(G\), if \(G=\bigcup^n_{i=1} H_i\) and \(H_i \cap H_j = T\) for every \(1 \leq i, j \leq n\). If \(\Pi\) is a strict \(T\)-partition of \(G\) and the orders of all components of \(\Pi\) are equal, then we say that \(G\) has an \(ET\)-partition. Here we show that: A finite group \(G\) is nilpotent if and only if every subgroup \(H\) of \(G\) has an ES-partition, for some \(S \leq H\).
MSC:
| 20D15 | Finite nilpotent groups, \(p\)-groups |
| 20E34 | General structure theorems for groups |
| 20D30 | Series and lattices of subgroups |
| 20D60 | Arithmetic and combinatorial problems involving abstract finite groups |
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