Commensurability in Group Theory
In group theory, a generalisation to pairs of subgroups is obtained, by noticing that in the case given, the subgroups of the integers as an additive group, generated respectively by a and by b, intersect in the subgroup generated by d, where d is the LCM of a and b. This intersection has finite index in the integers, and therefore in each of the subgroups. This gives rise to a general notion of commensurable subgroups: two subgroups A and B of a group are commensurable when their intersection has finite index in each of them. That is, two subgroups H1 and H2 of a group G are commensurable if
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The relation of being commensurable in the wide sense is that H1 be commensurable with a conjugate of H2. Some authors use the terms commensurate and commensurable for commensurable and widely commensurable respectively.
A relationship can similarly be defined on subspaces of a vector space, in terms of projections that have finite-dimensional kernel and cokernel.
In contrast, two subspaces and that are given by some moduli space stacks over a Lie algebra are not necessarily commensurable if they are described by infinite dimensional representations. In addition, if the completions of -type modules corresponding to and are not well-defined, then and are also not commensurable.
Read more about this topic: Commensurability (mathematics)
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