By Benjamin Fine

A survey of one-relator items of cyclics or teams with a unmarried defining relation, extending the algebraic research of Fuchsian teams to the extra basic context of one-relator items and comparable workforce theoretical issues. It presents a self-contained account of yes average generalizations of discrete teams.

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For, if G is a Cernikov group with ﬁnite residual D, then G embeds in the wreath product D wr (G/D). 3 The Cernikov groups are precisely the groups that arise as subgroups of standard wreath products DwrF where D is a direct product of ﬁnitely many quasicyclic groups and F is a ﬁnite group. ˇ Of course we get all the soluble Cernikov groups by taking F to be soluble. ˇ Clearly, to get further information about Cernikov groups, it is necessary to study modules over ﬁnite groups which are direct sums of ﬁnitely many quasicyclic ˇ groups: these have been called Cernikov modules.

We are now able to express the qS in terms of the pS by a recursive procedure. 4 p{α,β} = q{α} q{β} q{α,β} = p{α} p{β} q{α,β} . Hence −1 q{α,β} = p−1 {β} p{α} p{α,β} , and so on. Now specialize to the case where p = xr1 xr2 · · · xrn and label the factors as follows: (1) (2) (r) (1) (r) (r) x1 x1 · · · x1 x2 · · · x2 · · · x(1) n · · · xn , (i) so that xj = xj for 1 ≤ i ≤ r. If S ⊆ R with |S| = w, then w w pS = xw 1 x2 · · · xn , a product which depends only on |S|. 4 the element qS depends only on |S|, so that we may write qS = τw (x1 , x2 , .

20. But these factors are also ﬁnitely generated and so they are ﬁnite π-groups. Thus the group is a ﬁnite π-group. We note that by contrast the lower central series of a ﬁnitely generated torsion-free nilpotent group need not be torsion-free. For example, suppose n > 1 and set 1 an b G = 0 1 c | a, b, c ∈ Z . 0 0 1 Thus G is a torsion-free nilpotent subgroup of Un (Z). However, 1 0 dn G = 0 1 0 | d ∈ Z , 0 0 1 so that Gab ∼ = Z ⊕ Z ⊕ Zn . 3 Polycyclic groups The simplest type of abelian group is of course a cyclic group.