Let G1 and G2 be subgroups of a group G. A. Show that G1 intersect G2 is also a subgroup of G. B. Is G1 union G2 always a subgroup of G?

GATE 1995 · Discrete Mathematics · Group Theory · medium

Answer: G1 ∩ G2 is always a subgroup of G; G1 ∪ G2 is a subgroup only when one contains the other, so it is not always a subgroup.

1. Intersection is non-empty: Every subgroup contains the identity e, so e lies in both, hence in G1 ∩ G2.
2. Apply the subgroup test to the intersection: If a, b are in both subgroups then a * b^(-1) is in each (since each is a subgroup), so it is in the intersection.
3. Counterexample for the union: 5 is neither even nor a multiple of 3, so the union is not closed under +.