1.6.41 Fields, Groups, Rings

Condition: Let \( X \) be a subset of the group \( G \) such that \( X X \subseteq X \) (i.e. \( x y \in X ; \forall x, y \in X \) ). Prove that: 1) If \( X \) is finite or consists of periodic elements, then \( X \) is a subgroup of \( G \), 2) In the general case, \( X \) may not be a subgroup of \( G \).