Number of subgroups of direct product is bounded below by product of number of subgroups in each factor

From Groupprops
Jump to: navigation, search

Statement

Suppose G_1 and G_2 are groups. Then, the number of subgroups of the external direct product G_1 \times G_2 is at least equal to the product of the (number of subgroups of G_1) and the (number of subgroups of G_2):

(Number of subgroups of G_1 \times G_2) \ge (Number of subgroups of G_1)(Number of subgroups of G_2)

Proof

We will construct an injective set map:

(Set of subgroups of G_1) \times (Set of subgroups of G_2) \to (Set of subgroups of G_1 \times G_2)

The map is as follows: for subgroups H_1 \le G_1 and H_2 \le G_2, we have:

(H_1,H_2) \mapsto H_1 \times H_2