Volume 219, Nº 4 (2016)

The Hochschild cohomology groups are calculated for the algebras of semidihedral type that form the family \( SD\left(3\mathcal{K}\right) \) (from the famous K. Erdmann’s classification). In the calculation, the bimodule resolution previously constructed for the algebras belonging to the family under discussion is used.

A theorem on the stacked decomposition for infinitely generated projective left modules over serial left noetherian rings is proved.

For each prime p, the list of the sporadic simple groups and Suzuki groups whose p-modular group rings are serial is presented. Bibliography: 30 titles.

A finite group G is called a Schur group if every Schur ring over G is the transitivity module of a point stabilizer in a subgroup of Sym(G) that contains all permutations induced by the right multiplications in G. It is proved that the group \( {\mathrm{\mathbb{Z}}}_2\times {\mathrm{\mathbb{Z}}}_{2^n} \) is Schur, which completes the classification of Abelian Schur 2-groups. It is also proved that any non-Abelian Schur 2-group of order larger than 32 is dihedral (the Schur 2-groups of smaller orders are known). Finally, the Schur rings over a dihedral 2-group G are studied. It turns out that among such rings of rank at most 5, the only obstacle for G to be a Schur group is a hypothetical ring of rank 5 associated with a divisible difference set.

Let G = G(Φ,K) be a Chevalley group of type Φ over a field K, where Φ is a simply laced root system. By studying the extraspecial unipotent radical of G, it is proved that any its element is a product of at most three root elements. Moreover, it is shown that up to conjugation by an element of the Levi subgroup, any element of the radical is the product of six elementary root elements.