For any integer
n
≥
2
n \geq 2
, denote by
R
n
{R_n}
the class of groups
G
G
in which every infinite subset
X
X
contains
n
n
elements
x
1
,
…
,
x
n
{x_1}, \ldots ,{x_n}
such that the product
x
1
…
x
n
=
x
σ
(
1
)
⋯
x
σ
(
n
)
{x_1} \ldots {x_n} = {x_{\sigma (1)}} \cdots {x_{\sigma (n)}}
for some permutation
σ
≠
1
\sigma \ne 1
. The case
n
=
2
n = 2
was studied by B. H. Neumann who proved that
R
2
{R_2}
is precisely the class of centre-by-finite groups. Here we show that
G
∈
R
n
G \in {R_n}
for some
n
n
if and only if
G
G
contains an FC-subgroup
F
F
of finite index such that the exponent of
F
/
Z
(
F
)
F/Z(F)
is finite, where
Z
(
F
)
Z(F)
denotes the centre of
F
F
.