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

Let $H$ be a subgroup of $\text{Sym}_n$, the symmetric group of degree $n$. For a fixed integer $l \geq 2$, the group $G$ presented with generators $x_1, x_2, \ldots ,x_n$ and with relations $x_{i_1}x_{i_2}\cdots x_{i_l} =x_{\sigma (i_1)} x_{\sigma (i_2)} \cdots x_{\sigma (i_l)}$, 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 $\mathcal{J}(K[G])$ 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.