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The first part of the problem asks you to prove that an abelian group $G$ with order $100$ must contain an element of order $10$. For this part, I use Sylow theorem to list possiblities for $H$ and $K$ where $|H|$=$2^{2}$ and $|K|$=$5^{2}$. $K$ must be normal since there is not subgroup of order $25$ while $H$ might not be normal since $1$$\equiv$$25$ mod $2$. Also I proved that $H$$K$=$G$ and they have only the identity element in common.

Then, if $H$ and $K$ are normal. G must be isomorphic to one of the following groups:

$Z_{4}$ $\times$ $Z_{25}$

$Z_{2}$ $\times$ $Z_{2}$ $\times$ $Z_{25}$

$Z_{4}$ $\times$ $Z_{5}$ $\times$ $Z_{5}$

$Z_{2}$ $\times$ $Z_{2}$ $\times$ $Z_{5}$ $\times$ $Z_{5}$

From above, it is easily to pick up element of order $10$ for each. But my confusion is that since $G$ is an abelian group, how can I use the theorem that any finite abelian group is isomorphic to a direct product of cyclic groups?

Also, since the second part asks you if no element of $G$ has order greater than $10$, what are its torsion coefficients? I think my way of listing the possibilities are too complicated. Are there any more explicit ways to solve the problem?

Thanks a lot.

  • 0
    Is your overarching assumption that $G$ is abelian? Or do you need to prove $G$ is abelian after the first part of the problem?2012-05-10
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    @ArturoMagidin Sorry for the confusion, it is already given that $G$ is an abelian. That's why I am thinking of using the finite abelian group theorem but find the sylow theorem more easily to handle.2012-05-10
  • 0
    Gioven your group is commutative, can you find, by any means, an element of order 2, and an element of order 5? What are the possible orders of elements greater than 10 in an abelian group of order 100? Which orders must you have? How do you exclude these possibilities?2012-05-10
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    @Simonaster: I don't think you should _ever_ need Sylow's theorems when working with finite abelian groups. Note also that _every_ subgroup of an abelian group is normal.2012-05-10
  • 1
    You don't need the structure theorem of finitely generated abelian groups to solve this problem. It suffices to have Cauchy and Lagrange in hand.2012-05-10

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