Abstract

A group is called an \(n\)-torsion group if it has a system of defining relations of the form \(r^{n}=1\) for some elements \(r\), and for any of its finite order element \(a\) the defining relation \(a^{n}=1\) holds. In this paper, we prove that all the finite subgroups of the relatively free \(n\)-torsion groups are cyclic groups. Notice that for each rank \(m>1\) and for any odd \(n\geq 1003\), the set of nonisomorphic relatively free \(n\)-torsion groups of rank \(m\) has the cardinality of the continuum.

中文翻译：

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相对自由的$$ \ boldsymbol {n} $$ -Torsion组的有限子组

摘要

如果一个组具有为某些元素\（r \）及其任何有限的形式定义形式为\（r ^ {n} = 1 \）的关系的系统，则该组称为\（n \）-扭转组。顺序元素\（a \）定义关系\（a ^ {n} = 1 \）成立。在本文中，我们证明了相对自由的\（n \）扭转组的所有有限子组都是循环组。请注意，对于每个等级\（m> 1 \）和任何奇数\（n \ geq 1003 \），等级\（m \）的非同构相对自由\（n \）-扭转组的基数为连续体。