Suppose n is even positive integer and $H$ is a subgroup of $\mathbb Z_n$

Asked May 21 '16 at 06:31

Active May 21 '16 at 09:30

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Suppose $n$ is even positive integer and $H$ is a subgroup of $\mathbb Z_n$. Prove that either every member of $H$ is even or exactly half of members of $H$ are even.

Same question have been asked here, but I ask it again because I do not understand answers because of concepts like isomorphism and cosets etc. Need a simple answer.

Thanks

edited Apr 13 '17 at 12:19

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asked May 21 '16 at 06:31

Gathdi

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$H$ will have the form ${0,d,2d,...,(n/d-1)d}$ for some divisor $d | n.$ The two cases are whether $d$ is even or odd. – user341050 May 21 '16 at 06:39
If you don't like the answers given, the correct procedure is to make further comments etc there, not to ask the question again. – almagest May 21 '16 at 07:43
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Gathdi, I've answered it without using any "advanced" stuff. It can be made shorter and simpler if you first separately prove that every subgroup of a cyclic group is cyclic — the proof is very similar to a major part of the proof in my answer. – M. Vinay May 21 '16 at 14:54

Suppose n is even positive integer and $H$ is a subgroup of $\mathbb Z_n$

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