Abstract commensurability of groups 
and 
means that a finite-index subgroup of 
is isomorphic to a finite-index subgroup
of 
(Lück 2008). The subgroups need not lie in a common
ambient group. For example, 
and 
are abstractly commensurable. This is an equivalence
relation, since intersecting finite-index subgroups preserves finite index and allows the subgroup isomorphisms to be composed.
Abstract Commensurability
See also
Bifinite Correspondence, Equivalence Relation, Group, Isomorphic Groups, Subgroup, Subgroup IndexExplore with Wolfram|Alpha
References
Lück, W. "Survey on Geometric Group Theory." Münster J. Math. 1, 73-108, 2008. https://arxiv.org/abs/0806.3771.Cite this as:
Weisstein, Eric W. "Abstract Commensurability." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/AbstractCommensurability.html