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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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References:
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