On \(p\)-groups as loop groups
Let \(Q\) be a loop; then the mappings \(L_ a(x)=ax\) and \(R_ a(x)=xa\), where \(a\in Q\), are permutations of \(Q\), and they generate a permutation group \(M(Q)\), which is called the multiplication group of \(Q\). In the paper \(p\)-groups are considered as loop groups (a loop group is a group that is isomorphic to the multiplication group of a loop). The methods used rely mainly on the concept of connected transversals [see \textit{M. Niemenmaa} and \textit{T. Kepka}, J. Algebra 135, 112-122 (1990; Zbl 0706.20046)]. First, the non-commutative loop \(p\)-groups of minimal order are classified. If \(p=2\), then they are of order 32 and there exist four non-isomorphic such groups. If \(p\) is odd, then they are of order \(p^ 4\) and if \(p=3\), then there exists a unique such group, and if \(p>3\), then there exist two non-isomorphic non-commutative loop groups of order \(p^ 4\). After this, the Sylow \(p\)-subgroups of the symmetric groups \(S_{p^ n}\), \(p\) an odd prime, are shown to be loop groups. Finally, it is proved that a nilpotent group is a loop group if and only if its Sylow subgroups are loop groups.
non- commutative loop \(p\)-groups of minimal order
Sylow \(p\)-subgroups of symmetric groups
scientific article; zbMATH DE number 446286