# Lattice of subgroups embeds in partition lattice

## Statement

Suppose $G$ is a group, $P(G)$ is the partition lattice of $G$ (i.e., the lattice whose elements are partitions of $G$, with $\alpha \le \beta$ if $\alpha$ is a finer partition than $\beta$. Let $L(G)$ denote the Lattice of subgroups (?) of $G$.

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

1. For subgroups $H, K$ the left coset partition arising from the Join of subgroups (?) $\langle H, K \rangle$ is the same as the join in the partition lattice of the left coset partitions arising from $H$ and $K$ respectively.
2. For subgroups $H, K$, the left coset partition arising from the Intersection of subgroups (?) $H \cap K$ is the same as the meet in the partition lattice of the left coset partitions arising from $H$ and $K$ respectively.