CI-groups with respect to ternary relational structures: new examples

Abstract

We find a sufficient condition to establish that certain abelian groups are not CI-groups with respect to ternary relational structures, and then show that the groups Z3 × Z 2 2 , Z7 × Z 3 2 , and Z5 × Z 4 2 satisfy this condition. Then we completely determine which groups Z 3 2 × Zp, p a prime, are CI-groups with respect to binary and ternary relational structures. Finally, we show that Z 5 2 is not a CI-group with respect to ternary relational structures.

On the Isomorphism Problem for Cayley Graphs of Abelian Groups whose Sylow Subgroups are Elementary Abelian or Cyclic

“…There are now at least three obstacles to a group being CI with respect to digraphs that will depend at least to some extent on the structure of the full automorphism group of the digraph. The first is that there may not be an appropriate conjugate such that G L , δ −1 Gδ is normally m-step imprimitive [16]. Second, isomorphic regular abelian subgroups of a p-subgroup P of the symmetric group need not be conjugate in P .…”

“…The third is that in general the direct product of two CI-groups of relatively prime order need not be a CI-group by Theorem 1. All of these obstacles can occur in the automorphism group of a ternary relational structure ( [16] in the first instance, and [9] for the latter two). It thus seems wise to begin to break these large difficult problems into more manageable pieces.…”

“…is normally m-step imprimitive (this is not explicitly stated in [16,Theorem 2.1], but is established in the proof of that result). Recently, the author [13] extended the results in [8] to nilpotent groups, as well as to a larger class of abelian groups.…”

“…is nilpotent, in which case the Cayley isomorphism problem for G reduces to the Cayley isomorphism problem for the Sylow subgroups of G. It turns out that the "first part" of the above proofs does not generalize to arbitrary abelian groups of order n. The author and Spiga [16,Theorem 2.1] give examples of abelian groups of certain orders n and δ ∈ Sym(n) such that there is no…”

Enumerating Cayley (di-)graphs on dihedral groups

“…In the process, we also enumerate (connected) Cayley digraphs on D 2p of out-degree k up to isomorphism for each k. (Q. Huang).2 byÁdám [1]: all circulant graphs are CI-graphs of the corresponding cyclic groups. This conjecture was disproved by Elspas and Turner [9], and however, the conjecture caused a lot of activity on the characterization of CI-graphs (DCI-graphs) or CIgroups (DCI-groups) [3][4][5][6][7][8][11][12][13][14][15]17,[20][21][22]24]. Another motivation for investigating CI-graphs (DCI-graphs) or CI-groups (DCI-groups) is to determine and enumerate the isomorphic classes of Cayley graphs for a given group.…”