Cyclic groups are CI-groups for balanced configurations
In this paper it is shown that every finite cyclic group satisfies the CI-property for the class of balanced configurations.
math.CO↗
arXiv subjects
Mikhail Muzychuk
Publications and source records attributed to Mikhail Muzychuk.
The Schur-Wielandt theory for central S-rings
Two basic results on the S-rings over an abelian group are the Schur theorem on multipliers and the Wielandt theorem on primitive S-rings over groups with a cyclic Sylow subgroup. None of these theorems is directly generalized to the non-abelian case. Nevertheless, we prove that they are true for the central S-rings, i.e., for those which are contained in the center of the group ring of the underlying group (such S-rings naturally arise in the supercharacter theory). We also generalize the concept of a B-group introduced by Wielandt, and show that any Camina group is a generalized B-group whereas with few exceptions, no simple group is of this type.
math.GR↗
On Schur 2-groups
A finite group $G$ is called a Schur group, if any Schur ring over $G$ is the transitivity module of a point stabilizer in a subgroup of $\sym(G)$ that contains all right translations. We complete a classification of abelian $2$-groups by proving that the group $\mZ_2\times\mZ_{2^n}$ is Schur. We also prove that any non-abelian Schur $2$-group of order larger than $32$ is dihedral (the Schur $2$-groups of smaller orders are known). Finally, in the dihedral case, we study Schur rings of rank at most $5$, and show that the unique obstacle here is a hypothetical S-ring of rank $5$ associated with a divisible difference set.
math.CO↗
On Skew Hadamard difference sets
In this paper we construct exponentionally many non-isomorphic skew Hadamard difference sets over an elementary abelian group of order $q^3$.
math.CO↗
Symmetric Bush-type Hadamard matrices of order $4m^4$ exist for all odd $m$
Using reversible Hadamard difference sets, we construct symmetric Bush-type Hadamard matrices of order $4m^4$ for all odd integer $m$.
math.CO↗