### Radicals and Plotkin's problem concerning geometrically equivalent groups

by Goebel and Shelah. [GbSh:741]

Proc American Math Soc, 2002

If G and X are groups and N is a normal subgroup of X,
then the G- closure of N in X is the normal subgroup
{overline X}^G= bigcap {ker phi |phi :X-> G,
with N subseteq ker phi} of X . In particular,
{overline 1}^G = R_GX is the G- radical of X . Plotkin calls
two groups G and H geometrically equivalent, written G~H,
if for any free group F of finite rank and any normal subgroup N
of F the G --closure and the H --closure of N in F are the
same. Quasiidentities are formulas of the form (bigwedge_{i <= n}
w_i = 1-> w =1) for any words w, w_i (i <= n) in a
free group. Generally geometrically equivalent groups satisfy the
same quasiidentiies. Plotkin showed that nilpotent groups G and
H satisfy the same quasiidenties if and only if G and H are
geometrically equivalent. Hence he conjectured that this might hold
for any pair of groups. We provide a counterexample.

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