Finite \(p\)-groups all of whose proper subgroups have cyclic Frattini subgroups. (Q2903541)
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scientific article; zbMATH DE number 6064792
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Finite \(p\)-groups all of whose proper subgroups have cyclic Frattini subgroups. |
scientific article; zbMATH DE number 6064792 |
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10 August 2012
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finite \(p\)-groups
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cyclic Frattini subgroup
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0.9244087
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0.91500413
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0.91357887
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0.90781707
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0.9029025
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0.8882338
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Finite \(p\)-groups all of whose proper subgroups have cyclic Frattini subgroups. (English)
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A \(p\)-group \(G\) is said to be \(\Phi\mathcal C_p\)-group provided its Frattini subgroup is cyclic. In this paper the \(\Phi\mathcal C_p\)-groups for odd \(p\) are classified. The wider class of \(p\)-groups all of whose maximal subgroups have cyclic derived subgroups was treated in a paper by \textit{A. Leone}, [Matematiche 38(1987), 191-200 (1983; Zbl 0647.20013)] (see also \S139 of the book on prime power groups by the reviewer and \textit{Z. Janko} [Groups of prime power order. Vol. 3. Berlin: Walter de Gruyter (2011; Zbl 1229.20001)]). Exposition in this paper is independent of the above two sources.
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