Group algebras and semigroup algebras defined by permutation relations of fixed length

摘要

Let H be a subgroup of Sym(n), the symmetric group of degree n. For a fixed integer l >= 2, the group G presented with generators x(1), x(2),..., xn and with relations x(i1)x(i2) ... x(il) = x(sigma(i1)) xs((i2)) ... xs((il)), where sigma runs through H, is considered. It is shown that G has a free subgroup of finite index. For a field K, properties of the algebra K[G] are derived. In particular, the Jacobson radical J (K[G]) is always nilpotent, and in many cases the algebra K[G] is semiprimitive. Results on the growth and the Gelfand-Kirillov dimension of K[G] are given. Further properties of the semigroup S and the semigroup algebra K[S] with the same presentation are obtained, in case S is cancellative. The Jacobson radical is nilpotent in this case as well, and sufficient conditions for the algebra to be semiprimitive are given.