We investigate the discrete Fuglede's conjecture and Pompeiu problem on finite abelian groups and develop a strong connection between the two problems. We give a geometric condition under which a multiset of a finite abelian group has the discrete Pompeiu property. Using this description and the revealed connection we prove that Fuglede's conjecture holds for Z p n q 2 , where p and q are different primes. In particular, we show that every spectral subset of Z p n q 2 tiles the group. Further, using our combinatorial methods we give a simple proof for the statement that Fuglede's conjecture holds for Z 2 p .

For every prime p > 2 we exhibit a Cayley graph on Z 2p+3 p which is not a CI-graph. This proves that an elementary abelian p-group of rank greater than or equal to 2p + 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.

We investigate monotone idempotent n-ary semigroups and provide a generalization of the Czogala-Drewniak Theorem, which describes the idempotent monotone associative functions having a neutral element. We also present a complete characterization of idempotent monotone n-associative functions on an interval that have neutral elements.

An n-ary associative function is called reducible if it can be written as a composition of a binary associative function. We summarize known results when the function is defined on a chain and is nondecreasing. Our main result shows that associative idempotent and nondecreasing functions are uniquely reducible.