# Thread: subgroup question with direct product

1. ## subgroup question with direct product

Let G = Z+Z10 (+ is direct sum) and let H = { g an element of G | |g|= or |g|=1 }. Prove or disprove that H is a subgroup of G. [Subgroups that contain no elements of finite order besides the identity are called torsion-free subgroups. For example, Z+{0} is a torsion-free subgroup of G.] I really am stuck on this problem, if someone could help me with it, it would be great

2. Originally Posted by nhk
Let G = Z+Z10 (+ is direct sum) and let H = { g an element of G | |g|= or |g|=1 }. Prove or disprove that H is a subgroup of G. [Subgroups that contain no elements of finite order besides the identity are called torsion-free subgroups. For example, Z+{0} is a torsion-free subgroup of G.] I really am stuck on this problem, if someone could help me with it, it would be great

Hint: check the elements $(1,1),\,(-1,1)\in H$

Tonio

3. so since (1,1),(-1,1) are elements of H, then by closure (1,1)*(-1,1) must also be in H. But (1,1)*(-1,1)=(0,2) which has order 5. Thus H is not a subgroup of G.
Does that sound right?

4. Originally Posted by nhk
so since (1,1),(-1,1) are elements of H, then by closure (1,1)*(-1,1) must also be in H. But (1,1)*(-1,1)=(0,2) which has order 5. Thus H is not a subgroup of G.
Does that sound right?
Yes.