Question

Prove that an abelian group with two elements of order 2 must have a subgroup of order 4

Answer 1

Answer:

No group can have exactly two elements of order 2. Hint: Consider the two cases where ab = ba and ab ≠ba. Solution: Suppose that a and b are elements of order 2 in G. Then if ab = ba then, ab is a third element of order 2, while otherwise, aba is a third element of order 2.