Isomorphism Problem for Relational Structures with a Cyclic Automorphism

Classification of Cyclic Steiner Quadruple Systems

“…For example, Bays [3] and Lambossy [31] showed that if p is a prime then isomorphic cyclic Steiner systems S(t, k, p) are multiplier equivalent (a more general result was later obtained by Pálfy [35]). Examples of structures that are isomorphic but not multiplier equivalent have also been published: Brand [4] presented the first example of isomorphic cyclic Steiner systems that are not multiplier equivalent by finding such a family of S(2, 3, p n )s when p ≡ 1 (mod 6) is a prime and n ≥ 2 (a simpler construction was later given by Phelps [37]).…”

“…For completeness, these numbers are listed for all degrees up to 32. Notice that N is 0 exactly when the degree d = 4 or d and ϕ(d) are coprime [35]. 1 0 0 17 0 5 2 0 0 18 130 135 3 0 0 19 0 2 4 0 1 20 102 222 5 0 2 21 22 92 6 1 6 22 5 26 7 0 3 23 0 3 8 5 10 24 1359 2043 9 8 8 25 53 53 10 5 15 26 17 31 11 0 4 27 304 118 12 12 79 28 84 507 13 0 3 29 0 2 14 5 28 30 575 1290 15 0 70 31 0 4 16 179 115 32 26655 4341 If the automorphism group of an CSQS(v) is larger than its multiplier automorphism group, then the group has more than one regular cyclic subgroup.…”

The isomorphism problem for Cayley ternary relational structures for some abelian groups of order 8p

“…Replacing φ by φα for appropriate α ∈ Aut(Z 3 2 ), we may assume without loss of generality that E is formed by the orbits of ω . As Z 2 2 is a CI-group with respect to every class of combinatorial objects [17], after an appropriate conjugation, we may assume by Lemma 2.12 that G, φ −1 Gφ /E ∼ = Z 2 2 . Now, by Lemma 3.4 (with p = 2), we have that one of the following is true:…”

Galois correspondence between permutation groups and cellular rings (association schemes)

“…For other values of n counterexamples were found soon and the conjecture was transformed into the problem to find necessary and sufficient conditions for isomorphism of cyclic graphs on n vertices. The history of the problem of Adam can be traced by the survey paper [78].…”