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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:
 [1] Ádám, A., Research problem 2-10, J. combin. theory, 2, 393, (1967) [2] Alspach, B.; Parsons, T.D., Isomorphism of circulant graphs and digraphs, Discrete math., 25, 97-108, (1979) · Zbl 0402.05038 [3] Babai, L., Isomorphism problem for a class of point-symmetric structures, Acta math. acad. sci. hungar., 29, 329-336, (1977) · Zbl 0378.05035 [4] Babai, L.; Frankl, P., Isomorphism of Cayley graphs, I, Colloq. math. soc. J. Bólyai, (1978), North-Holland Amsterdam, p. 35-52 · Zbl 0404.05029 [5] Babai, L.; Frankl, P., Isomorphism of Cayley graphs, II, Acta math. acad. sci. hungar., 34, 177-183, (1979) · Zbl 0449.05029 [6] Bannai, E.; Song-Sung-Yell, Character tables of fission schemes, European J. combin., 14, 385-396, (1993) · Zbl 0794.05131 [7] Bannai, E.; Ito, T., Algebraic combinatorics. I. association schemes, (1984), Benjamin/Cummings Menlo Park · Zbl 0555.05019 [8] Conder, M.; Li, C.H., On isomorphisms of finite Cayley graphs, European J. of combin., 19, 911-919, (1998) · Zbl 0916.05034 [9] Djoković, D.Ź., Isomorphism problem for a special class of graphs, Acta math. acad. sci. hungar., 21, 267-270, (1970) [10] Dobson, E., Isomorphism problem for Cayley graphs of $$Z$$^3p, Discrete math., 147, 87-94, (1995) · Zbl 0838.05081 [11] Elspas, B.; Turner, J., Graphs with circulant adjacency matrices, J. combin. theory, 9, 297-307, (1970) · Zbl 0212.29602 [12] Godsil, C.D., On Cayley graphs isomorphism, Ars combin., 15, 231-246, (1983) · Zbl 0519.05036 [13] Higman, D.G., Coherent algebras, Linear algebra appl., 93, 209-239, (1987) · Zbl 0618.05014 [14] Klin, M.Ch.; Pöschel, R., The isomorphism problem for circulant graphs and digraphs with p^n vertices, Akad. der wiss. der DDR, (1980) · Zbl 0445.05050 [15] Klin, M.H.; Pöschel, R., The König problem, the isomorphism problem for cyclic graphs and the method of Schur rings, Colloq. math soc. János Bólyai, 25, 405-434, (1981) [16] Li, C.H.; Praeger, C.E.; Xu, M.Y., On finite groups with Cayley isomorphism property, J. graph theory, 27, 1-111, (1998) [17] Li, C.H., On isomorphism of connected Cayley graphs, Discrete math., 178, 109-122, (1998) · Zbl 0886.05078 [18] Li, C.H., The finite groups with the 2-DCI-property, Comm. algebra, 24, 1749-1757, (1996) · Zbl 0853.20003 [19] Li, C.H., On isomorphism of connected Cayley graphs, II, J. combin. theory ser. B, 74, 28-34, (1998) [20] Li, C.H., Finite CI-groups are soluble, Bull. London math. soc., 31, 419-423, (1999) · Zbl 0927.05036 [21] Li, C.H.; Praeger, C.E., On the isomorphism problem for finite Cayley graphs of bounded valency, European J. combin., 20, 279-292, (1999) · Zbl 0924.05035 [22] Muzychuk, M., Ádám’s conjecture is true in the square-free case, J. combin. theory ser. A, 72, 118-134, (1995) · Zbl 0833.05063 [23] Muzychuk, M., On ádám’s conjecture for circulant graphs, Discrete math., 176, 285-298, (1997) · Zbl 0885.05090 [24] M. Muzychuk, and, R. Pöschel, Isomorphism criterion for circulant graphs, manuscript. [25] Sabidussi, G., Vertex-transitve graphs, Monatsh. math., 68, 426-438, (1964) · Zbl 0136.44608 [26] Schur, I., Zür theorie der einfach transitiven permutationgruppen, S.-B. preus akad. wiss. phys.-math. kl., 598-623, (1933) · JFM 59.0151.01 [27] Tamaschke, O., A generalization of conjugacy in groups, Rend. sem. mat. univ. Padova, 40, 408-427, (1968) · Zbl 0174.04803 [28] Wielandt, H., Finite permutation groups, (1964), Academic Press Berlin · Zbl 0138.02501 [29] Wielandt, H., Permutation groups through invariant relations and invariant functions, Lecture notes, (1969) [30] Zieschang, P.H., An algebraic approach to association schemes, Lecture notes in math., 1628, (1996), Springer-Verlag New York/Berlin
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