Modular property of groups

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This result is related to the lattice of subgroups in a group

Statement

Symbolic statement

Let A, B and C be subgroups of a group G with the property that A \le C. Then:

A (B \cap C) = AB \cap C

Note here that AB denotes the product of subgroups and is not in general a group.

Implications

In case A commutes with the groups B and B \cap C, then the above can be recast as saying that the modular identity holds for the lattice of subgroups. This has the following easy implications: