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Levi quasivarieties of exponent $p^s$
V. V. Lodeishchikova Barnaul, Russia
Abstract:
For an arbitrary class
$M$ of groups,
$L(M)$ denotes a class of all groups
$G$ the normal closure of any element in which belongs to
$M$;
$qM$ is a quasivariety generated by
$M$. Fix a prime
$p$,
$p\ne2$, and a natural number
$s$,
$s\ge2$. Let
$qF$ be a quasivariety generated by a relatively free group in a class of nilpotent groups of class at most 2 and exponent
$p^s$, with commutator subgroups of exponent
$p$. We give a description of a Levi class generated by
$qF$.
Fix a natural number
$n$,
$n\ge2$. Let
$K$ be an arbitrary class of nilpotent groups of class at most
$2$ and exponent
$2^n$, with commutator subgroups of exponent
$2$. Assume also that for all groups in
$K$, elements of order
$2^m$,
$0<m<n$, are contained in the center of a given group. It is proved that a Levi class generated by a quasivariety
$qK$ coincides with a variety of nilpotent groups of class at most
$2$ and exponent
$2^n$, with commutator subgroups of exponent
$2$.
Keywords:
quasivariety, Levi classes, nilpotent groups.
UDC:
512.54.01 Received: 25.12.2009
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