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| Titel: |
A note on minimal subgroups of finite groups |
| Auteur: |
Asaad, M. Ballester-Bolinches, A. Aguilera, M.C. Pedraza |
| Verschenen in: |
Communications in algebra |
| Paginering: | Jaargang 24 (1996) nr. 8 pagina's 2771-2776 |
| Jaar: |
1996 |
| Inhoud: |
All groups considered in this note will be finite. Recall that a minimal subgroup of a finite group is a subgroup of prime order. Many authors have investigated the structure of a finite group G, under the assumption that all minimal subgroups of G are well-situated in the group. Ito [7;III, 5.3] proved that if G is a group of odd order and all minimal subgroups of G lie in the center of G, then G is nilpotent. An extension of Ito's result is the following statement [7;IV,p.435]: If for an odd prime p, every subgroup of G of order p lies in the center of G, then G is p-nilpotent. If all element of G of orders 2 and 4 lie in the center of G, then G is 2-nilpotent. Buckley [4] proved that if G is a group of odd order and all minimal subgroups of G are normal in G, then G is supersoluble. Later Shaalan [8] proved that if G is a finite group and every subgroup of G of prime order or order 4 is π-quasinormal in G, then G is supersoluble. Recall that a subgroup H of a group G is π-quasinormal in G if H permutes with every Sylow subgroup of G. More recently, Shirong [9] proved that if the finite group G possesses a normal subgroup N of odd order such that G/N is supersoluble, and if, for each Sylow subgroup P of N, every minimal subgroup of P is normal in NG(P), then G is supersoluble. In [10] and [11] Yokoyama extends the results of Ito and Buckley for soluble groups using formation theory to generalize the notion of centrality. |
| Uitgever: |
Taylor & Francis |
| Bronbestand: |
Elektronische Wetenschappelijke Tijdschriften |
Details van artikel 2 van 13 gevonden artikelen
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