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Commensurability (mathematics)
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==In group theory== {{main|Commensurability (group theory)}} In [[group theory]], two [[subgroup]]s Ξ<sub>1</sub> and Ξ<sub>2</sub> of a group ''G'' are said to be '''commensurable''' if the [[intersection (set theory)|intersection]] Ξ<sub>1</sub> β© Ξ<sub>2</sub> is of [[finite index]] in both Ξ<sub>1</sub> and Ξ<sub>2</sub>. Example: Let ''a'' and ''b'' be nonzero real numbers. Then the subgroup of the real numbers '''R''' [[generating set of a group|generated]] by ''a'' is commensurable with the subgroup generated by ''b'' if and only if the real numbers ''a'' and ''b'' are commensurable, in the sense that ''a''/''b'' is rational. Thus the group-theoretic notion of commensurability generalizes the concept for real numbers. There is a similar notion for two groups which are not given as subgroups of the same group. Two groups ''G''<sub>1</sub> and ''G''<sub>2</sub> are ('''abstractly''') '''commensurable''' if there are subgroups ''H''<sub>1</sub> β ''G''<sub>1</sub> and ''H''<sub>2</sub> β ''G''<sub>2</sub> of finite index such that ''H''<sub>1</sub> is [[group isomorphism|isomorphic]] to ''H''<sub>2</sub>.
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