On generalized smooth groups. (English) Zbl 1116.20015
The theory of smooth groups has enjoyed a rapid evolution in the last years. Many other classes of groups connected with smooth groups were also studied. We mention here the well-known papers of R. Schmidt or H. Schnabel.
In the present paper there are investigated the generalized smooth groups, namely the groups \(G\) which satisfy the property that the lattice interval \([G/H]\) is totally smooth, for every subgroup \(H\) of \(G\). The structure of such a group is presented in the Main Theorem. The author determines five classes of finite generalized smooth groups.
All the results of the paper are new, interesting and carefully proved, and so they contribute to the development of this research domain.
In the present paper there are investigated the generalized smooth groups, namely the groups \(G\) which satisfy the property that the lattice interval \([G/H]\) is totally smooth, for every subgroup \(H\) of \(G\). The structure of such a group is presented in the Main Theorem. The author determines five classes of finite generalized smooth groups.
All the results of the paper are new, interesting and carefully proved, and so they contribute to the development of this research domain.
Reviewer: Marius Tarnauceanu (Iaşi)
MSC:
| 20D30 | Series and lattices of subgroups |
Keywords:
lengths of intervals; maximal chains; generalized smooth groups; subgroup lattices of finite groupsReferences:
| [1] | DOI: 10.1023/A:1010333719254 · Zbl 0985.20012 · doi:10.1023/A:1010333719254 |
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