Selected exercises from Abstract Algebra by Dummit and Foote (3rd edition)

We conclude thhat T is a subgroup. Now we exhibit an example where T fails to be a subgroup when G is non-abelian. Let G = 〈x, y|x = y = i 〉. Observe that the product xy has infinite order since the product (xy)(xy) is no longer reducible because we lack the commutativity of x with y. Hence, x, y ∈ T but (xy) / ∈ T . Exercise (7). Fix some n ∈ Z with n > 1. Find the torsion subgroup of Z×(Z/nZ). Show that the set of elements of infinite order together with the identity is not a subgroup of this direct product.