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Section4.1Cyclic Subgroups

Ā¶

Often a subgroup will depend entirely on a single element of the group; that is, knowing that particular element will allow us to compute any other element in the subgroup.

Example4.1

Suppose that we consider $3 \in {\mathbb Z}$ and look at all multiples (both positive and negative) of 3. As a set, this is

\begin{equation*} 3 {\mathbb Z} = \{ \ldots, -3, 0, 3, 6, \ldots \}. \end{equation*}

It is easy to see that $3 {\mathbb Z}$ is a subgroup of the integers. This subgroup is completely determined by the element 3 since we can obtain all of the other elements of the group by taking multiples of 3. Every element in the subgroup is ā€œgeneratedā€ by 3.

Example4.2

If $H = \{ 2^n : n \in {\mathbb Z} \}\text{,}$ then $H$ is a subgroup of the multiplicative group of nonzero rational numbers, ${\mathbb Q}^*\text{.}$ If $a = 2^m$ and $b = 2^n$ are in $H\text{,}$ then $ab^{-1} = 2^m 2^{-n} = 2^{m-n}$ is also in $H\text{.}$ By PropositionĀ 3.31, $H$ is a subgroup of ${\mathbb Q}^*$ determined by the element 2.

Theorem4.3

Let $G$ be a group and $a$ be any element in $G\text{.}$ Then the set

\begin{equation*} \langle a \rangle = \{ a^k : k \in {\mathbb Z} \} \end{equation*}

is a subgroup of $G\text{.}$ Furthermore, $\langle a \rangle$ is the smallest subgroup of $G$ that contains $a\text{.}$

Proof

The identity is in $\langle a \rangle $ since $a^0 = e\text{.}$ If $g$ and $h$ are any two elements in $\langle a \rangle \text{,}$ then by the definition of $\langle a \rangle$ we can write $g = a^m$ and $h = a^n$ for some integers $m$ and $n\text{.}$ So $gh = a^m a^n = a^{m+n}$ is again in $\langle a \rangle \text{.}$ Finally, if $g = a^n$ in $\langle a \rangle \text{,}$ then the inverse $g^{-1} = a^{-n}$ is also in $\langle a \rangle \text{.}$ Clearly, any subgroup $H$ of $G$ containing $a$ must contain all the powers of $a$ by closure; hence, $H$ contains $\langle a \rangle \text{.}$ Therefore, $\langle a \rangle $ is the smallest subgroup of $G$ containing $a\text{.}$

Remark4.4

If we are using the ā€œ+ā€ notation, as in the case of the integers under addition, we write $\langle a \rangle = \{ na : n \in {\mathbb Z} \}\text{.}$

For $a \in G\text{,}$ we call $\langle a \rangle $ the generated by $a\text{.}$ If $G$ contains some element $a$ such that $G = \langle a \rangle \text{,}$ then $G$ is a . In this case $a$ is a of $G\text{.}$ If $a$ is an element of a group $G\text{,}$ we define the of $a$ to be the smallest positive integer $n$ such that $a^n= e\text{,}$ and we write $|a| = n\text{.}$ If there is no such integer $n\text{,}$ we say that the order of $a$ is infinite and write $|a| = \infty$ to denote the order of $a\text{.}$

Example4.5

Notice that a cyclic group can have more than a single generator. Both 1 and 5 generate ${\mathbb Z}_6\text{;}$ hence, ${\mathbb Z}_6$ is a cyclic group. Not every element in a cyclic group is necessarily a generator of the group. The order of $2 \in {\mathbb Z}_6$ is 3. The cyclic subgroup generated by 2 is $\langle 2 \rangle = \{ 0, 2, 4 \}\text{.}$

The groups ${\mathbb Z}$ and ${\mathbb Z}_n$ are cyclic groups. The elements 1 and $-1$ are generators for ${\mathbb Z}\text{.}$ We can certainly generate ${\mathbb Z}_n$ with 1 although there may be other generators of ${\mathbb Z}_n\text{,}$ as in the case of ${\mathbb Z}_6\text{.}$

Example4.6

The group of units, $U(9)\text{,}$ in ${\mathbb Z}_9$ is a cyclic group. As a set, $U(9)$ is $\{ 1, 2, 4, 5, 7, 8 \}\text{.}$ The element 2 is a generator for $U(9)$ since

\begin{align*} 2^1 & = 2 \qquad 2^2 = 4\\ 2^3 & = 8 \qquad 2^4 = 7\\ 2^5 & = 5 \qquad 2^6 = 1. \end{align*}
Example4.7

Not every group is a cyclic group. Consider the symmetry group of an equilateral triangle $S_3\text{.}$ The multiplication table for this group is TableĀ 3.7. The subgroups of $S_3$ are shown in FigureĀ 4.8. Notice that every subgroup is cyclic; however, no single element generates the entire group.

Figure4.8Subgroups of $S_3$
Theorem4.9

Every cyclic group is abelian.

Proof

Let $G$ be a cyclic group and $a \in G$ be a generator for $G\text{.}$ If $g$ and $h$ are in $G\text{,}$ then they can be written as powers of $a\text{,}$ say $g = a^r$ and $h = a^s\text{.}$ Since

\begin{equation*} g h = a^r a^s = a^{r+s} = a^{s+r} = a^s a^r = h g, \end{equation*}

$G$ is abelian.

SubsectionSubgroups of Cyclic Groups

Ā¶

We can ask some interesting questions about cyclic subgroups of a group and subgroups of a cyclic group. If $G$ is a group, which subgroups of $G$ are cyclic? If $G$ is a cyclic group, what type of subgroups does $G$ possess?

Theorem4.10

Every subgroup of a cyclic group is cyclic.

Proof

The main tools used in this proof are the division algorithm and the Principle of Well-Ordering. Let $G$ be a cyclic group generated by $a$ and suppose that $H$ is a subgroup of $G\text{.}$ If $H = \{ e \}\text{,}$ then trivially $H$ is cyclic. Suppose that $H$ contains some other element $g$ distinct from the identity. Then $g$ can be written as $a^n$ for some integer $n\text{.}$ Since $H$ is a subgroup, $g^{-1} = a^{-n}$ must also be in $H\text{.}$ Since either $n$ or $-n$ is positive, we can assume that $H$ contains positive powers of $a$ and $n \gt 0\text{.}$ Let $m$ be the smallest natural number such that $a^m \in H\text{.}$ Such an $m$ exists by the Principle of Well-Ordering.

We claim that $h = a^m$ is a generator for $H\text{.}$ We must show that every $h' \in H$ can be written as a power of $h\text{.}$ Since $h' \in H$ and $H$ is a subgroup of $G\text{,}$ $h' = a^k$ for some integer $k\text{.}$ Using the division algorithm, we can find numbers $q$ and $r$ such that $k = mq +r$ where $0 \leq r \lt m\text{;}$ hence,

\begin{equation*} a^k = a^{mq +r} = (a^m)^q a^r = h^q a^r. \end{equation*}

So $a^r = a^k h^{-q}\text{.}$ Since $a^k$ and $h^{-q}$ are in $H\text{,}$ $a^r$ must also be in $H\text{.}$ However, $m$ was the smallest positive number such that $a^m$ was in $H\text{;}$ consequently, $r=0$ and so $k=mq\text{.}$ Therefore,

\begin{equation*} h' = a^k = a^{mq} = h^q \end{equation*}

and $H$ is generated by $h\text{.}$

Corollary4.11

The subgroups of ${\mathbb Z}$ are exactly $n{\mathbb Z}$ for $n = 0, 1, 2,\ldots\text{.}$

Proposition4.12

Let $G$ be a cyclic group of order $n$ and suppose that $a$ is a generator for $G\text{.}$ Then $a^k=e$ if and only if $n$ divides $k\text{.}$

Proof

First suppose that $a^k=e\text{.}$ By the division algorithm, $k = nq + r$ where $0 \leq r \lt n\text{;}$ hence,

\begin{equation*} e = a^k = a^{nq + r} = a^{nq} a^r = e a^r = a^r. \end{equation*}

Since the smallest positive integer $m$ such that $a^m = e$ is $n\text{,}$ $r= 0\text{.}$

Conversely, if $n$ divides $k\text{,}$ then $k=ns$ for some integer $s\text{.}$ Consequently,

\begin{equation*} a^k = a^{ns} = (a^n)^s = e^s = e. \end{equation*}
Theorem4.13

Let $G$ be a cyclic group of order $n$ and suppose that $a \in G$ is a generator of the group. If $b = a^k\text{,}$ then the order of $b$ is $n/d\text{,}$ where $d = \gcd(k,n)\text{.}$

Proof

We wish to find the smallest integer $m$ such that $e = b^m = a^{km}\text{.}$ By PropositionĀ 4.12, this is the smallest integer $m$ such that $n$ divides $km$ or, equivalently, $n/d$ divides $m(k/d)\text{.}$ Since $d$ is the greatest common divisor of $n$ and $k\text{,}$ $n/d$ and $k/d$ are relatively prime. Hence, for $n/d$ to divide $m(k/d)$ it must divide $m\text{.}$ The smallest such $m$ is $n/d\text{.}$

Corollary4.14

The generators of ${\mathbb Z}_n$ are the integers $r$ such that $1 \leq r \lt n$ and $\gcd(r,n) = 1\text{.}$

Example4.15

Let us examine the group ${\mathbb Z}_{16}\text{.}$ The numbers 1, 3, 5, 7, 9, 11, 13, and 15 are the elements of ${\mathbb Z}_{16}$ that are relatively prime to 16. Each of these elements generates ${\mathbb Z}_{16}\text{.}$ For example,

\begin{align*} 1 \cdot 9 & = 9 & 2 \cdot 9 & = 2 & 3 \cdot 9 & = 11\\ 4 \cdot 9 & = 4 & 5 \cdot 9 & = 13 & 6 \cdot 9 & = 6 \\ 7 \cdot 9 & = 15 & 8 \cdot 9 & = 8 & 9 \cdot 9 & = 1 \\ 10 \cdot 9 & = 10 & 11 \cdot 9 & = 3 & 12 \cdot 9 & = 12\\ 13 \cdot 9 & = 5 & 14 \cdot 9 & = 14 & 15 \cdot 9 & = 7. \end{align*}