show that N(G) is a lattice?

Question

if $G$ is a group

let N(G) be the set of normal subgroups of $G$ define $\wedge$ and $\vee$ on$ N(G)$ by $N_{1}\wedge N_{2}=N_{1}\cap N_{2}$ and $N_{1}\vee N_{2}=N_{1} N_{2}=\{n_{1}n_{2}:n_{1}\in N_{1} , n_{2}\in N_{2}\}$

show that N(G) is a lattice? answer:

commutative laws and associative laws is hold but for idempotent laws and absorption how to continue?

Answer 1

Hint: Idempotency for $\cap$ is obvious, and for the join $\vee$ just use the fact that you are working with subgroups (closed under product).

The absorption laws will follow from the fact that both subgroups contain the group identity.