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Let $C_n$ be the cyclic group of order $n$. I'm trying to investigate properties of

$\bigcup_{n=1}^{\infty} C_n$

It seems obvious that this is a group, but I don't really know much else about it. Does this group have a name, and are there any references where properties of this group are described?

Edit: The group operation here is given by thinking of the elements of $C_n$ as complex nth roots of unity. For any two elements $x \in C_n$ and $y \in C_m$, the product $xy \in C_{mn}$ is just given by complex multiplication.

Mark B
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    If we think of $C_n$ as the complex $n^{th}$ roots of unity, then the group operation is just multiplication of complex numbers. Then something in $C_3$ multiplied by something in $C_5$ will (in general) be in $C_{15}$. – Mark B Nov 11 '14 at 02:08
  • Edited the question to include the group multiplication. – Mark B Nov 11 '14 at 02:14
  • Consider the set of rational numbers with addition mod 1. This is your group. – MJD Nov 11 '14 at 02:50

1 Answers1

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You're probably taking the direct limit with the usual inclusions $C_d\hookrightarrow C_n$ (for $d\mid n$). To compute the direct limit of $\Bbb Z/n\Bbb Z$, think of it instead as $(\frac{1}{n}\Bbb Z)/\Bbb Z$ (as additive subgroups of $\Bbb Q$), so that your union is easily seen to be $\bigcup (\frac{1}{n}\Bbb Z/\Bbb Z)=(\bigcup \frac{1}{n}\Bbb Z)/\Bbb Z=\Bbb Q/\Bbb Z$. A direct limit of a chain of cyclic groups is called a locally cyclic group; these all embed into $\Bbb Q/\Bbb Z$ or $\Bbb Q$ (the first in the case of a torsion group, the second in the case of a torsion-free group). Another definition of locally cyclic is when all finitely-generated subgroups are cyclic. We know that $\Bbb Q/\Bbb Z$ decomposes as a direct sum of its $p$-primary components, which are called the Prüfer $p$-groups ${\Bbb Z}(p^\infty)\cong\Bbb Z[1/p]/\Bbb Z$.

There is a general trick for computing basic direct limits of chains linked to and illustrated here.

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