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An elementary Abelian group of rank 4 is a CI-group. (English) Zbl 1002.20030
Finite groups are dealt with. In addition to the known notion of CI-group (i.e., group possessing the Cayley isomorphism property), the authors consider the related concept of \(\text{CI}^{(2)}\)-group.
Let \(F\), \(G\) be subgroups of the symmetric (permutation) group \(\text{Sym}(X)\). We say that \(G(\supseteq F)\) is \(F\)-transjugate if \(G\) acts transitively on all its subgroups which are conjugate to \(F\) in \(\text{Sym}(X)\). For an arbitrary group \(H\), let \(H_R\) be the subgroup of \(\text{Sym}(H)\) consisting of all right multiplications by the elements of \(H\). \(H\) is called a \(\text{CI}^{(2)}\)-group if every 2-closed overgroup of \(H_R\) is \(H_R\)-transjugate. {For the definition of 2-closedness, see e.g. Section 8.1 of the book of L. A. Kaluzhnin and R. Pöschel [Funktionen- und Relationsalgebren, Deutscher Verlag d. Wiss., Berlin (1979; Zbl 0418.03044)].}
The main results of the article assert that \(\mathbb{Z}^4_p\) is a CI-group for every prime \(p\) (this fact was already known in case \(p=2\)), and \(\mathbb{Z}^m_p\) is a \(CI^{(2)}\)-group if \(m\leq 4\) and \(p\) is an arbitrary odd prime. The proof of these theorems is achieved by lengthy considerations using Schur rings and their isomorphisms. – Any finite \(\text{CI}^{(2)}\)-group is clearly a CI-group, the converse statement is an open question.

MSC:
20K01 Finite abelian groups
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
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