The construction of infinite finitely generated periodic groups using wreath products
Let n be a natural number, \(n\geq 4\), and let \(\pi\) be the set of the prime divisors of n. With the help of generalized wreath products, an infinite periodic \(\pi\)-group \(E_ n\) is constructed with two generators of order n and with the following additional properties: (i) some member of the transfinite derived series of \(E_ n\) is equal to the identity, (ii) the group \(E_ n\) is residually finite, (iii) \(Z(E_ n)=1\), (iv) \(E_ n\) contains an isomorphic copy of every countable residually finite locally soluble FC-\(\pi\)-group. Furthermore, if n divides m then the groups \(E_ n\) and \(E_ m\) may be constructed such that \(E_ n\leq E_ m\). It is to be noted that an example of a 2-generated periodic group of Aleshin type containing an element of arbitrary finite order was constructed by \textit{A. V. Rozhkov} [Mat. Zametki 40, No.5, 572-589 (1986; Zbl 0614.20019)].
residually finite locally soluble FC-\(\pi \)-group
scientific article; zbMATH DE number 4044090
scientific article