In this paper, we show that the group Zp5 is a DCI-group for any odd prime p, that is, two Cayley digraphs Cay(Zp5,S) and Cay(Zp5,T) are isomorphic if and only if S=Tφ for some automorphism φ of the ...group Zp5.
We describe two similar but independently-coded computations used to construct a complete catalogue of the transitive groups of degree less than 48, thereby verifying, unifying and extending the ...catalogues previously available. From this list, we construct all the vertex-transitive graphs of order less than 48. We then present a variety of summary data regarding the transitive groups and vertex-transitive graphs, focusing on properties that seem to occur most frequently in the study of groups acting on graphs. We illustrate how such catalogues can be used, first by finding a complete list of the elusive groups of order at most 47 and then by completely determining which groups of order at most 47 are CI groups.
A Cayley (di)graph
of a group G is called normal if the right regular representation of G is normal in the full automorphism group of
and a CI-(di)graph if for every Cayley (di)graph
implies that ...there is
such that
We call a group G an NDCI-group or NCI-group if all normal Cayley digraphs or graphs of G are CI-digraphs or CI-graphs, respectively. We prove that a cyclic group of order n is an NDCI-group if and only if
and an NCI-group if and only if either n = 8 or
A group $G$ is a CI-group with respect to graphs if two Cayley graphs of $G$ are isomorphic if and only if they are isomorphic by a group automorphism of $G$. We show that an infinite family of ...groups which include $D_n\times F_{3p}$ are not CI-groups with respect to graphs, where $p$ is prime, $n\not = 10$ is relatively prime to $3p$, $D_n$ is the dihedral group of order $n$, and $F_{3p}$ is the nonabelian group of order $3p$.
A tournament is an oriented complete graph. The problem of ranking tournaments was firstly investigated by P. Erdős and J. W. Moon. By probabilistic methods, the existence of ˆˆ ˆˆ unrankable” ...tournaments was proved. On the other hand, they also mentioned the problem of explicit constructions. However, there seems to be only a few of explicit constructions of such tournaments. In this note, we give a construction of many such tournaments by using skew Hadamard difference sets which have been investigated in combinatorial design theory.
We show that if certain arithmetic conditions hold, then the Cayley isomorphism problem for abelian groups, all of whose Sylow subgroups are elementary abelian or cyclic, reduces to the Cayley ...isomorphism problem for its Sylow subgroups. This yields a large number of results concerning the Cayley isomorphism problem, perhaps the most interesting of which is the following: if $p_1,\ldots, p_r$ are distinct primes satisfying certain arithmetic conditions, then two Cayley digraphs of $\mathbb{Z}_{p_1}^{a_1}\times\cdots\times\mathbb{Z}_{p_r}^{a_r}$, $a_i\le 5$, are isomorphic if and only if they are isomorphic by a group automorphism of $\mathbb{Z}_{p_1}^{a_1}\times\cdots\times\mathbb{Z}_{p_r}^{a_r}$. That is, that such groups are CI-groups with respect to digraphs.
A finite group R is a DCI-group if, whenever S and T are subsets of R with the Cayley graphs Cay(R,S) and Cay(R,T) isomorphic, there exists an automorphism φ of R with Sφ=T.
Elementary abelian groups ...of order p4 or smaller are known to be DCI-groups, while those of sufficiently large rank are known not to be DCI-groups. The only published proof that elementary abelian groups of order p4 are DCI-groups uses Schur rings and does not work for p=2 (which has been separately proven using computers). This paper provides a simpler proof that works for all primes. Some of the results in this paper also apply to elementary abelian groups of higher rank, so may be useful for completing our determination of which elementary abelian groups are DCI-groups.
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