Abstract. Maximal connected grading classes of a finite-dimensional algebra A are in one-to-one correspondence with Galois covering classes of A which admit no proper Galois covering and therefore are key in computing the intrinsic fundamental group π 1 (A). Our first concern here is the algebras A = Mn(C). Their maximal connected gradings turn out to be in one-to-one correspondence with the Aut(G)-orbits of non-degenerate classes in H 2 (G, C * ), where G runs over all groups of central type whose orders divide n 2 . We show that there exist groups of central type G such that H 2 (G, C * ) admits more than one such orbit of non-degenerate classes. We compute the family Λ of positive integers n such that there is a unique group of central type of order n 2 , namely Cn × Cn. The family Λ is of square-free integers and contains all prime numbers. It is obtained by a full description of all groups of central type whose orders are cube-free. We establish the maximal connected gradings of all finite dimensional semisimple complex algebras using the fact that such gradings are determined by dimensions of complex projective representations of finite groups. In some cases we give a description of the corresponding fundamental groups.

We prove that the Herzog-Schönheim Conjecture holds for any group G of order smaller than 1440. In other words we show that in any non-trivial coset partitionWe also study interaction between the indices of subgroups having cosets with pairwise trivial intersection and harmonic integers. We prove that if U 1 ,...,Un are subgroups of G which have pairwise trivially intersecting cosets and n ≤ 4 then [G : U 1 ],...,[G : Un] are harmonic integers.Theorem A. Any group of order less than 1440 satisfies the Herzog-Schönheim Conjecture.

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Similarly to how the classical group ring isomorphism problem asks, for a commutative ring R, which information about a finite group G is encoded in the group ring RG, the twisted group ring isomorphism problem asks which information about G is encoded in all the twisted group rings of G over R.We investigate this problem over fields. We start with abelian groups and show how the results depend on the characteristic of R. In order to deal with non-abelian groups we construct a generalization of a Schur cover which exists also when R is not an algebraically closed field, but still linearizes all projective representations of a group. We then show that groups from the celebrated example of Everett Dade which have isomorphic group algebras over any field can be distinguished by their twisted group algebras over finite fields.

When one studies the structure (e.g. graded ideals, graded subspaces, radicals,. . . ) or graded polynomial identities of graded algebras, the grading group itself does not play an important role, but can be replaced by any other group that realizes the same grading. Here we come to the notion of weak equivalence of gradings: two gradings are weakly equivalent if there exists an isomorphism between the graded algebras that maps each graded component onto a graded component. The following question arises naturally: when a group grading on a finite dimensional algebra is weakly equivalent to a grading by a finite group? It turns out that this question can be reformulated purely group theoretically in terms of the universal group of the grading. Namely, a grading is weakly equivalent to a grading by a finite group if and only if the universal group of the grading is residually finite with respect to a special subset of the grading group. The same is true for all the coarsenings of the grading if and only if the universal group of the grading is hereditarily residually finite with respect to the same subset. We show that if n 349, then on the full matrix algebra M n (F ) there exists an elementary group grading that is not weakly equivalent to any grading by a finite (semi)group, and if n 3, then any elementary grading on M n (F ) is weakly equivalent to an elementary grading by a finite group.be two gradings where S and T are (semi)groups and A and B are algebras.The most restrictive case is when we require that both grading (semi)groups coincide: Definition 1.1 (e.g. [11, Definition 1.15]). The gradings (1.1) are isomorphic if S = T and there exists an isomorphism ϕ : A → B of algebras such that ϕ(A (s) ) = B (s) for all s ∈ S. In this case we say that A and B are graded isomorphic. In some cases, such as in [16], less restrictive requirements are more suitable. Definition 1.2 ([16, Definition 2.3]). The gradings (1.1) are equivalent if there exists an isomorphism ϕ : A → B of algebras and an isomorphism ψ : S → T of (semi)groups such that ϕ(A (s) ) = B ψ(s) for all s ∈ S. Remark 1.3. The notion of graded equivalence was considered by Yu. A. Bahturin, S. K. Seghal, and M. V. Zaicev in [6, Remark after Definition 3]. In the paper of V. Mazorchuk and K. Zhao [19] it appears under the name of graded isomorphism. A. Elduque