  In 2001 Ivanov and Kerov associated with the infinite permutation group
$S_\infty$ certain commutative associative algebra $A_\infty$ called the
algebra of conjugacy classes of partial elements. A standard basis of
$A_\infty$ is labeled by Yang diagrams of all orders.
  Mironov, Morozov, Natanzon, 2012, have proved that the completion of
$A_\infty$ is isomorphic to the direct product of centers of group algebras of
groups $S_n$. This isomorphism was explored in a construction of infinite
dimensional Cardy-Frobenius algebra corresponding to asymptotic Hurwitz
numbers.
  In this work algebras of conjugacy classes of partial elements are defined
for a wider class of infinite groups. It is proven that completion of any such
algebra is isomorphic to the direct product of centers of group algebras of
relevant subgroups.
