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 casesthe algebra $K[G]$ is semiprimitive. Results on the growth and theGelfand-Kirillov dimension of $K[G]$ are given. Further properties of thesemigroup $S$ and the semigroup algebra $K[S]$ with the same presentation areobtained, in case $S$ is cancellative. The Jacobson radical is nilpotent inthis case as well, and sufficient conditions for the algebra to besemiprimitive are given.
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机译：假设$ H $是$ \ text {Sym} _n $的子组，即度数$ n $的对称组。对于固定整数$ l \ geq 2 $，$ G $组与生成器$ x_1，x_2一起出现， \ ldots，x_n $并具有关系$ x_ {i_1} x_ {i_2} \ cdots x_ {i_l} = x _ {\ sigma（i_1）} x _ {\ sigma（i_2）} \ cdots x _ {\ sigma（i_l）} $，其中$ \ sigma $贯穿$ H $。结果表明，$ G $具有一个自由的有限索引子组。对于字段$ K $，可以导出代数$ K [G] $的属性。特别是，Jacobson根$ \ mathcal {J}（K [G]）$始终是幂等的，在许多情况下，代数$ K [G] $是半本原的。给出了$ K [G] $的增长和Gelfand-Kirillov维度的结果。如果$ S $是可取消的，则获得具有相同表示的这些mi群$ S $和半群代数$ K [S] $的其他性质。 Jacobson根在这种情况下也是幂等的，并且给出了代数为半本原的充分条件。
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