# subgroup question with direct product

• April 5th 2010, 11:40 AM
nhk
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(Rofl)
• April 5th 2010, 12:13 PM
tonio
Quote:

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(Rofl)

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

Tonio
• April 6th 2010, 12:57 PM
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?
• April 6th 2010, 01:40 PM
Drexel28
Quote:

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.