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Math Help - subgroup question with direct product

  1. #1
    nhk
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    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
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    Quote Originally Posted by nhk View Post
    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
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  3. #3
    nhk
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    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?
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by nhk View Post
    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.
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