Elementary abelian p-groups of rank 2 p+3 are not CI-groups
Gábor Somlai
DOI: 10.1007/s10801-011-0273-9
Abstract
For every prime p>2 we exhibit a Cayley graph on \mathbb Z p 2 p+3 \mathbb{Z}_{p}^{2p+3} which is not a CI-graph. This proves that an elementary abelian p-group of rank greater than or equal to 2 p+3 is not a CI-group. The proof is elementary and uses only multivariate polynomials and basic tools of linear algebra. Moreover, we apply our technique to give a uniform explanation for the recent works of Muzychuk and Spiga concerning the problem.
Pages: 323–335
Keywords: keywords Cayley graph; CI-group; elementary abelian $p$-group
Full Text: PDF
References
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1976. Combinatorics, vol. 18, pp. 35-52. North-Holland, Amsterdam (1978)
2. Conder, M., Li, C.H.: On isomorphism of Cayley graphs. Eur. J. Comb. 19, 911-919 (1998)
3. Hirasaka, M., Muzychuk, M.: An elementary abelian group of rank 4 is a CI-group. J. Comb. Theory, Ser. A 94(2), 339-362 (2001)
4. Morris, J.: Results towards showing Z2p - 1 p is a CI-group. In: Proceedings of the Thirty-third Southeastern International Conference on Combinatorics, Graph Theory and Computing, Boca Raton, FL,
2002. Congr. Numer, vol. 156, pp. 143-153 (2002)
5. Muzychuk, M.: An elementary abelian group of large rank is not a CI-group. Discrete Math. 264(1-3), 167-185 (2003)
6. Nowitz, L.A.: A non-Cayley-invariant Cayley graph of the elementary abelian group of order
64. Discrete Math. 110, 223-228 (1992)
7. Spiga, P.: Elementary abelian p-groups of rank greater than or equal to 4p - 2 are not CI-groups.
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