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.]