4: Subgroups

From now on, unless otherwise stated, $G$ will denote a group whose binary operation is denoted by $a \cdot b$ or simply $ab$ for $a,b \in G$. The identity of $G$ will be denoted by $e$ and the inverse of $a \in G$ will be denoted by $a^{-1}$. Sometimes, however, we may need to discuss groups whose operations are thought of as addition. In such cases we write $a + b$ instead of $ab$. Also in this case, the identity is denoted by $0$ and the inverse of $a \in G$ is denoted by $-a$. Definitions and results given using multiplicative notation can always be translated to additive notation if necessary.

Definition 4.1:

Let $G$ be a group. A subgroup of $G$ is a subset $H$ of $G$ which satisfies the following three conditions:

1. $e \in H.$
2. If $a,b \in H$, then $ab \in H$.
3. If $a \in H$, then $a^{-1} \in H$.

For convenience we sometimes write $H \le G$ to mean that $H$ is a subgroup of $G$.

Problem 4.1 Translate the above definition into additive notation.

Remark

If $H$ is a subgroup of $G$, then the binary operation on $G$ when restricted to $H$ is a binary operation on $H$. From the definition, one may easily show that a subgroup $H$ is a group in its own right with respect to this binary operation. Many examples of groups may be obtained in this way. In fact, in a way we will make precise later, every finite group may be thought of as a subgroup of one of the groups $S_n$.

Problem 4.2 Prove that if $G$ is any group, then

1. $\{e \} \le G$.
2. $G \le G$.

The subgroups $\{e \}$ and $G$ are said to be trivial subgroups of $G$.

Problem 4.3

(a) Determine which of the following subsets of $S_4$ are subgroups of $S_4$.

1. $H=\{ \iota, (1 \ 2), (3 \ 4), (1 \ 2) (3 \ 4) \}$
2. $K=\{ \iota, (1 \ 2 \ 3), ( 1 \ 3 \ 2) \}$
3. $J=\{ \iota, (1 \ 2), (1 \ 2 \ 3) \}$
4. $L=\{\sigma\in S_4 \ | \ \sigma(1) = 1\}$.

(b) Determine which of the following subsets of $\mathbb{Z}_{12}$ are subgroups of $\mathbb{Z}_{12}$. (Here the binary operation is addition modulo 12.)

1. $A=\{ 0, 3, 6, 9, \}$
2. $B=\{ 0, 6 \}$
3. $C=\{0, 1,2,3,4,5 \}$

(c) Determine which of the following subsets of $\mathbb{Z}$ are subgroups of $\mathbb{Z}$. (Here the binary operation is addition.)

1. $U=\{ 5k \ \vert k \ \in \mathbb{Z}\}$
2. $V=\{ 5k + 1 \ \vert \ k \in \mathbb{N}\}$
3. $W=\{ 5k +1 \ \vert \ k \in \mathbb{Z}\}$

Problem 4.4 Let $$SL(2,\mathbb{R}) = \{ A \in GL(2,\mathbb{R}) \, \vert \, \det(A) = 1 \}.$$ Prove that $SL(2,\mathbb{R}) \le GL(2,\mathbb{R})$.

$SL(2,\mathbb{R})$ is called the Special Linear Group of Degree 2 over $\mathbb{R}$

Problem 4.5 For $n \in \mathbb{N}$, let $A_n$ be the set of all even permutations in the group $S_n$. Show that $A_n$ is a subgroup of $S_n$.

$A_n$ is called the alternating group of degree $n$.

Problem 4.6 List the elements of $A_n$ for $n = 1, 2, 3, 4$. Based on this try to guess the order of $A_n$ for $n > 4$.

Definition 4.2:

Let $a$ be an element of the group $G$. If there exists $n \in \mathbb{N}$ such that $a^n = e$ we say that $a$ has finite order. and we define $$\mathrm{o}(a) = \min \{ n \in \mathbb{N} \, | \, a^n=e \}$$ If $a^n \ne e$ for all $n \in \mathbb{N}$, we say that $a$ has infinite order and we define $$\mathrm{o}(a)=\infty.$$ In either case we call $\mathrm{o}(a)$ the order of $a$.

Note carefully the difference between the order of a group and the order of an element of a group. Some authors make matters worse by using the same notation for both concepts. Maybe by using different notation it will make it a little easier to distinguish the two concepts.

If $n \ge 2$, to prove that $\mathrm{o}(a)=n$ we must show that $a^i \ne e$ for $i=1, 2, \dots, n-1$ and $a^n =e$. Note also that $a=e$ if and only if $\mathrm{o}(a)=1$. So every element of a group other than $e$ has order $n \ge 2$ or $\infty$.

Problem 4.7 Translate the above definition into additive notation. That is, define the order of an element of a group $G$ with binary operation $+$ and identity denoted by 0.

Problem 4.8 Find the order of each element of $S_3$.

Problem 4.9 Find the order of a $k$-cycle when $k=2,3,4,5$. Guess the order of a $k$-cycle for arbitrary $k$.

Problem 4.10 Find the order of the following permutations:

(a) $(1 \ 2) (3 \ 4 \ 5)$

(b) $(1 \ 2) ( 3 \ 4) (5 \ 6 \ 7 \ 8)$

(c) $(1 \ 2) ( 3 \ 4) (5 \ 6 \ 7 \ 8) ( 9 \ 10 \ 11)$

(d) Try to find a rule for computing the order of a product disjoint cycles in terms of the sizes of the cycles.

Problem 4.11 Find the order of each element of the group $(\mathbb{Z}_6,+)$.

Problem 4.12 Find the order of each element of $GL(2,\mathbb{Z}_2)$. [Recall that $GL(2,\mathbb{Z}_2)$ is the group of all $2 \times 2$ matrices with entries in $Z_2$ with non-zero determinant. Recall that $\mathbb{Z}_2 = \{ 0, 1 \}$ and the operations are multiplication and addition modulo 2.]

Problem 4.13 Find the order of the element 2 in the group $(\mathbb{R}-\{ 0 \},\cdot)$. Are there any elements of finite order in this group?

Definition 4.3:

Let $a$ be an element of the group $G$. Define $$\langle a \rangle = \{ a^i: i \in \mathbb{Z}\}.$$ We call $\langle a \rangle$ the subgroup of $G$ generated by $a$.

Remark

Note that $$\langle a \rangle = \{ \ldots, a^{-3}, a^{-2}, a^{-1}, a^0, a^1, a^2, a^3, \ldots \}.$$ In particular, $a = a^1$ and $e = a^0$ are in $\langle a \rangle$.

Problem 4.14 Translate the above definition of $\langle a \rangle$ and the remark into additive notation.

Theorem $\PageIndex{1}$

For each $a$ in the group $G$, $\langle a \rangle$ is a subgroup of $G$. $\langle a \rangle$ contains $a$ and is the smallest subgroup of $G$ that contains $a$. $\blacksquare$

Proof As just noted $e=a^0 \in \langle a \rangle$. Let $a^n,a^m \in \langle a \rangle$. Since $n+m \in \mathbb{Z}$ it follows from Theorem 2.4 that $$a^na^m = a^{n+m} \in \langle a \rangle.$$ Also from Theorem 2.4, if $a^n \in \langle a \rangle$, since $n(-1) = -n$ we have $$(a^n)^{-1} = a^{-n} \in \langle a \rangle.$$ This proves that $\langle a \rangle$ is a subgroup.

Since $a=a^1$ it is clear that $a \in \langle a \rangle$. If $H$ is any subgroup of $G$ that contains $a$, since $H$ is closed under taking products and taking inverses, $a^n \in \langle a \rangle$ for every $n \in \mathbb{Z}$. So $\langle a \rangle \subseteq H$. That is, every subgroup of $G$ that contains $a$ also contains $\langle a \rangle$. This implies that $\langle a \rangle$ is the smallest subgroup of $G$ that contains $a$.

Theorem $\PageIndex{2}$

Let $G$ be a group and let $a \in G$. If $\mathrm{o}(a)=1$, then $\langle a \rangle = \{e\}$. If $\mathrm{o}(a) = n$ where $n \ge 2$, then $$\langle a \rangle = \{ e, a, a^2, \ldots, a^{n-1} \}$$ and the elements $e, a, a^2, \ldots, a^{n-1}$ are distinct, that is, $$\mathrm{o}(a) = \vert \langle a \rangle \vert.$$ $\blacksquare$

Proof Assume that $\mathrm{o}(a) = n$. The case $n=1$ is left to the reader. Suppose $n \ge 2$. We must prove two things.

1. If $i \in \mathbb{Z}$ then $a^i \in \{ e, a, a^2, \ldots, a^{n-1} \}$.
2. The elements $e, a, a^2, \ldots, a^{n-1}$ are distinct.

To establish 1 we note that if $i$ is any integer we can write it in the form $i=nq+r$ where $r \in \{ 0, 1, \dots, n-1 \}$. Here $q$ is the quotient and $r$ is the remainder when $i$ is divided by $n$. Now using Theorem [Th2.3] we have $$a^i=a^{nq+r}=a^{nq}a^r=(a^n)^qa^r = e^qa^r=ea^r=a^r.$$ This proves 1. To prove 2, assume that $a^i = a^j$ where $0 \le i < j \le n-1$. It follows that $$a^{j-i} = a^{j + (-i)} = a^ja^{-i} = a^ia^{-i}=a^0=e.$$ But $j-i$ is a positive integer less than $n$, so $a^{j-i} =e$ contradicts the fact that $\textrm{o}(a) =n$. So the assumption that $a^i = a^j$ where $0 \le i < j \le n-1$ is false. This implies that 2 holds. It follows that $\langle a \rangle$ contains exactly $n$ elements, that is, $\mathrm{o}(a) = \vert \langle a \rangle \vert.$

Theorem $\PageIndex{3}$

If $G$ is a finite group, then every element of $G$ has finite order. $\blacksquare$

Proof Let $a$ be any element of $G$. Consider the infinite list $$a^1, a^2, a^3, \dots, a^i, \dots$$ of elements in $G$. Since $G$ is finite, all the elements in the list cannot be different. So there must be positive integers $i < j$ such that $a^i = a^j$. Since $i < j$, $j-i$ is a positive integer. Then using Theorem 2.4 we have $$a^{j-i} = a^{j + (-i)} = a^ja^{-i} = a^ia^{-i}=a^0=e.$$ That is, $a^n = e$ for the positive integer $n=j-i$. So $a$ has finite order, which is what we wanted to prove.

Problem 4.15 For each choice of $G$ and each given $a \in G$ list all the elements of the subgroup $\langle a \rangle$ of $G$.

1. $G=S_3$, $a=( 1 \ 2)$.
2. $G=S_3$, $a=( 1 \ 2 \ 3 )$.
3. $G=S_4$, $a= ( 1 \ 2 \ 3 \ 4)$.
4. $G=S_4$, $a=( 1 \ 2)( 3 \ 4)$.
5. $G=\mathbb{Z}$, $a=5$.
6. $G=\mathbb{Z}$, $a=-1$.
7. $G=\mathbb{Z}_{15}$, $a=5$.
8. $G=\mathbb{Z}_{15}$, $a=1$.
9. $G=GL(2,\mathbb{Z}_2)$, $a= \left ( \begin{array} {cc} 1&1\\0&1 \end{array}\right)$.
10. $G=GL(2,\mathbb{R})$, $a= \left ( \begin{array} {cr} 0&-1 \\ 1&0 \end{array}\right)$.

Problem 4.16 Suppose $a$ is an element of a group and $o(a)= n$. Prove that $a^m = e$ if and only if $n \, \vert \, m$. [Hint: The Division Algorithm from Appendix C may be useful for the proof in one direction.]