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We study p-adic multiresolution analyses (MRAs). A complete characterization of test functions generating an MRA (scaling functions) is given. We prove that only 1-periodic test functions may be taken as orthogonal scaling functions and that all such scaling functions generate the Haar MRA. We also suggest a method for constructing sets of wavelet functions and prove that any set of wavelet functions generates a p-adic wavelet frame.

The generalized wreath product of permutation groups is introduced. By means of it we study the schurity problem for S-rings over a cyclic group G and the automorphism groups of them. Criteria for the schurity and non-schurity of the generalized wreath product of two such S-rings are obtained. As a byproduct of the developed theory we prove that G is a Schur group whenever the total number Ω(n) of prime factors of the integer n = |G| is at most 3. Moreover, we describe the structure of a non-schurian S-ring over G when Ω(n) = 4. The latter result implies in particular that if n = p 3 q where p and q are primes, then G is a Schur group.

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We define and study $m$-closed cellular algebras (coherent configurations) and $m$-isomorphisms of cellular algebras which can be regarded as $m$th approximations of Schurian algebras (i.e. the centralizer algebras of permutation groups) and of strong isomorphisms (i.e. bijections of the point sets taking one algebra to the other) respectively. If $m=1$ we come to arbitrary cellular algebras and their weak isomorphisms (i.e. matrix algebra isomorphisms preserving the Hadamard multiplication). On the other hand, the algebras which are $m$-closed for all $m\ge 1$ are exactly Schurian ones whereas the weak isomorphisms which are $m$-isomorphisms for all $m\ge 1$ are exactly ones induced by strong isomorphisms. We show that for any $m$ there exist $m$-closed algebras on $O(m)$ points which are not Schurian and $m$-isomorphisms of cellular algebras on $O(m)$ points which are not induced by strong isomorphisms. This enables us to find for any $m$ an edge colored graph with $O(m)$ vertices satisfying the $m$-vertex condition and having non-Schurian adjacency algebra. On the other hand, we rediscover and explain from the algebraic point of view the Cai-Fürer-Immerman phenomenon that the $m$-dimensional Weisfeiler-Lehman method fails to recognize the isomorphism of graphs in an efficient way.