# Lattice of subgroups embeds in partition lattice

From Groupprops

## Statement

Suppose is a group, is the partition lattice of (i.e., the lattice whose elements are partitions of , with if is a finer partition than . Let denote the Lattice of subgroups (?) of .

Then, there is an injective homomorphism of lattices from to , given as follows: a subgroup of is sent to the partition of given by the Left coset (?)s of in (as per left cosets partition a group). In particular:

- For subgroups the left coset partition arising from the Join of subgroups (?) is the same as the join in the partition lattice of the left coset partitions arising from and respectively.
- For subgroups , the left coset partition arising from the Intersection of subgroups (?) is the same as the meet in the partition lattice of the left coset partitions arising from and respectively.