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There is an exercise in the book "An Introduction to the group theory by J.J. Rose" which can also be found as a proposition in "Abstract algebra by T. Hungerford":

Every finite group has a composition series $^*$.

Now I am doing the exercise $5.9$ of the first above book:

  1. An abelian group has a composition series iff it is finite.

  2. Give an example of an infinite group which has a composition series.

About 1. : Since $(*)$; one side can be carried out. For other side; what would be happened if we assumed the group was infinite? In fact, if an abelian group is infinite; it cannot have a composition series with finite length? Is this our contradiction? I see this by considering $\mathbb Z_{p^\infty}$ but cannot see the right way. Thanks.

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    @BabakSorouh You can deduce that $G$ must be finitely generated and apply the fundamental theorem.2012-09-02

2 Answers 2

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A non-trivial simple Abelian group is cyclic of prime order. A composition series must have finite length. This should suffice.

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    Thanks for your kind help about 2. Thanks for that again. I think it would be a bit difficult to me to find it.2012-09-02
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I think you can proceed with your problem by reducing to the case for (1) that your abelian group must be finitely generated. For suppose that your abelian group $G$ has a composition series. Then it would follow (I think) that all chains in $G$ are bounded in length, consequently $G$ satisfies the ascending chain condition and descending chain condition (as a $\Bbb{Z}$ - module) and so is finitely generated . Now by the fundamental theorem of finitely generated abelian groups, we get that

$G \cong \Bbb{Z}^n \oplus \Bbb{Z}_{p_1} \oplus \ldots \oplus \Bbb{Z}_{p_n}$

for some prime numbers $p_1,\ldots,p_n$. Now if $n > 0$, you have a copy of $\Bbb{Z}$ sitting inside of $G$ that gives rise to a descending chain of subgroups inside of $G$ that does not terminate, contradicting $G$ being Artinian. It follows $n =0$ and consequently $G$ is a finite abelian group. $\hspace{6in} \square$

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    @BenjaL$i$m: Honestly, I was thinking about the fundamental theorem of finitely generated abelian groups but I wonder why J.J.Rose brought this question in the chapter 5 while he would discuss about abelian groups in chapter 10!!! :-)2012-09-02