Mixed identities and mixed varieties of groups

V. S. Anashin

Sbornik. Mathematics, Tome 57 (1987) no. 1, pp. 171-182

Résumé

A mixed identity in variables $x_1,x_2,\dots$ over a group $G$ is a word $g_1x_{i_1}^{m_1}\cdots g_kx_{i_k}^{m_k}g_{k+1}$ (where the coefficients $g_1,\dots,g_{k+1}$ lie in $G$, $i_1,\dots,i_k\in\{1,2,\dots\}$, and $m_1,\dots,m_k\in\mathbf Z$) taking the value 1 for any values of the variables in $G$. The concept of a mixed variety of groups is introduced as an object corresponding to a certain set of mixed identities and generalizing the concept of a variety of groups; an analogue of Birkhoff's theorem is proved; minimal mixed varieties generated by a finite group are described; the question of whether the mixed identities of a group can be derived from its identities is studied; and for nilpotent and metabelian groups it is established that all their mixed identities with coefficients in a finitely generated subgroup are finitely based, from which the same property is deduced for the identities of such groups with finitely many distinguished points. Bibliography: 16 titles.