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Lemma $1.92$ in Rotman's textbook (Advanced Modern Algebra, second edition) states,

Let $G = \langle a \rangle$ be a cyclic group.

(i) Every subgroup $S$ of $G$ is cyclic.

(ii) If $|G|=n$, then $G$ has a unique subgroup of order $d$ for each divisor $d$ of $n$.

I understand how every subgroup must be cyclic and that there must be a subgroup for each divisor of $d$. But how is that subgroup unique? I'm having trouble understanding this intuitively. For example, if we look at the cyclic subgroup $\Bbb{7}$, we know that there are $6$ elements of order $7$. So we have six different cyclic subgroups of order $7$, right?

Thanks in advance.

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Keith Conrad has some wonderful notes on this very topic. See Theorem 3.6 for your specific question. math.uconn.edu/~kconrad/blurbs/grouptheory/cyclicgp.pdf –  AWertheim Jun 3 '13 at 18:32
This article might help. See "Subgroups of cyclic groups" that relate to the one you are talking about. Hope this helps. –  NasuSama Jun 3 '13 at 18:33
@Artus, I will point out that you have some mistakes in your thinking. By "The cyclic subgroup 7", I assume you mean a cyclic group of order 7, say, the additive group $\mathbb{Z}/7\mathbb{Z}$. In this case, take two nonzero elements, say $1$, and $2$. Write out by hand the subgroups that these two elements generate. Are they different? –  AWertheim Jun 3 '13 at 18:36
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5 Answers

up vote 4 down vote accepted

To help you understand where you're going wrong, why not try writing out these "six different subgroups": if $G$ is a cyclic group of order $7$, and $a$ is a generator of $G$, then

$$\begin{array}{c|c} \mathsf{\text{Subgroup of }}G\mathsf{\text{ generated by}} & \mathsf{\text{consists of}}\\\hline a \strut & a,\; a^2,\;a^3,\; a^4,\; a^5,\;a^6,\;a^7=e\\\hline a^2 \strut& \\\hline \vdots \strut&\\\hline a^6\strut & \\\hline \end{array}$$

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+1 for a concrete way to see the error. –  Austin Mohr Jun 3 '13 at 18:44
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Suppose $\langle a \rangle$ has order $n$. If $d \mid n$, then $a^{n/d}$ has order $d$. Any subgroup of $\langle a \rangle$ is of the form $\langle a^k \rangle$ for some $k \mid n$. Hence if a subgroup has order $d \mid n$, it must be $\langle a^{n/d} \rangle$.

You are right that there are $6$ elements of order $7$ in a cyclic group of order $7$, but these all generate the same cyclic subgroup.

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I do not understand the "hence" here. –  user1729 Jul 27 '13 at 19:58
@user1729: Suppose that $H$ is a subgroup of order $d$, where $d \mid n$. Now $H = \langle a^k \rangle$ for some $k \mid n$. Then $a^k$ has order $n/k$, but on the other hand $a^k$ must also have order $d$. Hence $k = n/d$. –  Mikko Korhonen Jul 28 '13 at 17:36
Ah, right, I missed the $k\mid n$ bit. Thanks. –  user1729 Jul 28 '13 at 17:46
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Let $d$ be a divisor of $n=|G|$. Consider $H=\{ x \in G : x^d =1 \}$. Then $H$ is a subgroup of $G$ and $H$ contains all elements of $G$ that have order $d$ (among others).

If $K$ is a subgroup of $G$ of order $d$, then $K$ is cyclic, generated by an element of order $d$. Hence, $K\subseteq H$.

On the other hand, $x\in H$ iff $x=g^k$ with $0\le k < n$ and $g^{kd}=1$, where $g$ is a generator of $G$. Hence, $kd=nt$ and so $k=(n/d) t$. The restriction $0\le k<n$ implies $0\le t<d$, and so $H$ has exactly $d$ elements. Therefore, $K=H$.

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I like this proof a lot. Just thought I'd say :) –  user1729 Jul 27 '13 at 19:59
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Hint $\ $ The key idea for $\,\Bbb Z/n\,$ is the same as the proof for $\,\Bbb Z\!:\,$ since a subgroup S is closed under subtraction, every element of S is a multiple of the least element $> 0$ (using representatives $\ge 0$).

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Suppose $|G| = n$ and $G = \langle a \rangle = \{e,a,\ldots,a^{n-1}\}$. If $d$ divides $n$, then the subgroup $\langle a^{n/d} \rangle$ has order $d$. Conversely, suppose that $H = \langle g \rangle$ is a subgroup of order $d$ (we know it's cyclic by the first part). Now we know that $g = a^{k}$ for some $1 \leq k \leq n-1$. If $H$ has order $d$, can $k$ be anything other than $n/d$?

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Yes it can. Any cyclic group of order $> 2$ has more than one generator. –  Mikko Korhonen Jun 3 '13 at 18:44
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