Skip to main content

Section 4.1 Definition and Examples

Two groups \((G, \cdot)\) and \((H, \circ)\) are isomorphic if there exists a one-to-one and onto map \(\phi : G \rightarrow H\) such that the group operation is preserved; that is,

\begin{equation*} \phi( a \cdot b) = \phi( a) \circ \phi( b) \end{equation*}

for all \(a\) and \(b\) in \(G\text{.}\) If \(G\) is isomorphic to \(H\text{,}\) we write \(G \cong H\text{.}\) The map \(\phi\) is called an isomorphism.

Example 4.1.

To show that \({\mathbb Z}_4 \cong \langle i \rangle\text{,}\) define a map \(\phi: {\mathbb Z}_4 \rightarrow \langle i \rangle\) by \(\phi(n) = i^n\text{.}\) We must show that \(\phi\) is bijective and preserves the group operation. The map \(\phi\) is one-to-one and onto because

\begin{align*} \phi(0) & = 1\\ \phi(1) & = i\\ \phi(2) & = -1\\ \phi(3) & = -i\text{.} \end{align*}

Since

\begin{equation*} \phi(m + n) = i^{m+n} = i^m i^n = \phi(m) \phi( n)\text{,} \end{equation*}

the group operation is preserved.

Example 4.2.

We can define an isomorphism \(\phi\) from the additive group of real numbers \(( {\mathbb R}, + )\) to the multiplicative group of positive real numbers \(( {\mathbb R^+}, \cdot )\) with the exponential map; that is,

\begin{equation*} \phi( x + y) = e^{x + y} = e^x e^y = \phi( x ) \phi( y)\text{.} \end{equation*}

Of course, we must still show that \(\phi\) is one-to-one and onto, but this can be determined using calculus.

Example 4.3.

The integers are isomorphic to the subgroup of \({\mathbb Q}^\ast\) consisting of elements of the form \(2^n\text{.}\) Define a map \(\phi: {\mathbb Z} \rightarrow {\mathbb Q}^\ast\) by \(\phi( n ) = 2^n\text{.}\) Then

\begin{equation*} \phi( m + n ) = 2^{m + n} = 2^m 2^n = \phi( m ) \phi( n )\text{.} \end{equation*}

By definition the map \(\phi\) is onto the subset \(\{2^n :n \in {\mathbb Z} \}\) of \({\mathbb Q}^\ast\text{.}\) To show that the map is injective, assume that \(m \neq n\text{.}\) If we can show that \(\phi(m) \neq \phi(n)\text{,}\) then we are done. Suppose that \(m \gt n\) and assume that \(\phi(m) = \phi(n)\text{.}\) Then \(2^m = 2^n\) or \(2^{m - n} = 1\text{,}\) which is impossible since \(m - n \gt 0\text{.}\)

Example 4.4.

The groups \({\mathbb Z}_8\) and \({\mathbb Z}_{12}\) cannot be isomorphic since they have different orders; however, it is true that \(U(8) \cong U(12)\text{.}\) We know that

\begin{align*} U(8) & = \{1, 3, 5, 7 \}\\ U(12) & = \{1, 5, 7, 11 \}\text{.} \end{align*}

An isomorphism \(\phi : U(8) \rightarrow U(12)\) is then given by

\begin{align*} 1 & \mapsto 1\\ 3 & \mapsto 5\\ 5 & \mapsto 7\\ 7 & \mapsto 11\text{.} \end{align*}

The map \(\phi\) is not the only possible isomorphism between these two groups. We could define another isomorphism \(\psi\) by \(\psi(1) = 1\text{,}\) \(\psi(3) = 11\text{,}\) \(\psi(5) = 5\text{,}\) \(\psi(7) = 7\text{.}\) In fact, both of these groups are isomorphic to \({\mathbb Z}_2 \times {\mathbb Z}_2\) (see Example 2.28 in Chapter 2).

Example 4.5.

Even though \(S_3\) and \({\mathbb Z}_6\) possess the same number of elements, we would suspect that they are not isomorphic, because \({\mathbb Z}_6\) is abelian and \(S_3\) is nonabelian. To demonstrate that this is indeed the case, suppose that \(\phi : {\mathbb Z}_6 \rightarrow S_3\) is an isomorphism. Let \(a , b \in S_3\) be two elements such that \(ab \neq ba\text{.}\) Since \(\phi\) is an isomorphism, there exist elements \(m\) and \(n\) in \({\mathbb Z}_6\) such that

\begin{equation*} \phi( m ) = a \quad \text{and} \quad \phi( n ) = b\text{.} \end{equation*}

However,

\begin{equation*} ab = \phi(m ) \phi(n) = \phi(m + n) = \phi(n + m) = \phi(n ) \phi(m) = ba\text{,} \end{equation*}

which contradicts the fact that \(a\) and \(b\) do not commute.

Proof.

Assertions (1) and (2) follow from the fact that \(\phi\) is a bijection. We will prove (3) here and leave the remainder of the theorem to be proved in the exercises.

(3) Suppose that \(h_1\) and \(h_2\) are elements of \(H\text{.}\) Since \(\phi\) is onto, there exist elements \(g_1, g_2 \in G\) such that \(\phi(g_1) = h_1\) and \(\phi(g_2) = h_2\text{.}\) Therefore,

\begin{equation*} h_1 h_2 = \phi(g_1) \phi(g_2) = \phi(g_1 g_2) = \phi(g_2 g_1) = \phi(g_2) \phi(g_1) = h_2 h_1\text{.} \end{equation*}

We are now in a position to characterize all cyclic groups.

Proof.

Let \(G\) be a cyclic group with infinite order and suppose that \(a\) is a generator of \(G\text{.}\) Define a map \(\phi : {\mathbb Z} \rightarrow G\) by \(\phi : n \mapsto a^n\text{.}\) Then

\begin{equation*} \phi( m+n ) = a^{m+n} = a^m a^n = \phi( m ) \phi( n )\text{.} \end{equation*}

To show that \(\phi\) is injective, suppose that \(m\) and \(n\) are two elements in \({\mathbb Z}\text{,}\) where \(m \neq n\text{.}\) We can assume that \(m \gt n\text{.}\) We must show that \(a^m \neq a^n\text{.}\) Let us suppose the contrary; that is, \(a^m = a^n\text{.}\) In this case \(a^{m - n} = e\text{,}\) where \(m - n \gt 0\text{,}\) which contradicts the fact that \(a\) has infinite order. Our map is onto since any element in \(G\) can be written as \(a^n\) for some integer \(n\) and \(\phi(n) = a^n\text{.}\)

Proof.

Let \(G\) be a cyclic group of order \(n\) generated by \(a\) and define a map \(\phi : {\mathbb Z}_n \rightarrow G\) by \(\phi : k \mapsto a^k\text{,}\) where \(0 \leq k \lt n\text{.}\) The proof that \(\phi\) is an isomorphism is one of the end-of-chapter exercises.

The main goal in group theory is to classify all groups; however, it makes sense to consider two groups to be the same if they are isomorphic. We state this result in the following theorem, whose proof is left as an exercise.

Hence, we can modify our goal of classifying all groups to classifying all groups up to isomorphism; that is, we will consider two groups to be the same if they are isomorphic.

Subsection 4.1.1 Cayley's Theorem

Cayley proved that if \(G\) is a group, it is isomorphic to a group of permutations on some set; hence, every group is a permutation group. Cayley's Theorem is what we call a representation theorem. The aim of representation theory is to find an isomorphism of some group \(G\) that we wish to study into a group that we know a great deal about, such as a group of permutations or matrices.

Example 4.11.

Consider the group \({\mathbb Z}_3\text{.}\) The Cayley table for \({\mathbb Z}_3\) is as follows.

\begin{equation*} \begin{array}{c|ccc} + & 0 & 1 & 2 \\ \hline 0 & 0 & 1 & 2 \\ 1 & 1 & 2 & 0 \\ 2 & 2 & 0 & 1 \end{array} \end{equation*}

The addition table of \({\mathbb Z}_3\) suggests that it is the same as the permutation group \(G = \{ (0), (0 1 2), (0 2 1) \}\text{.}\) The isomorphism here is

\begin{align*} 0 & \mapsto \begin{pmatrix} 0 & 1 & 2 \\ 0 & 1 & 2 \end{pmatrix} = (0)\\ 1 & \mapsto \begin{pmatrix} 0 & 1 & 2 \\ 1 & 2 & 0 \end{pmatrix} = (0 1 2)\\ 2 & \mapsto \begin{pmatrix} 0 & 1 & 2 \\ 2 & 0 & 1 \end{pmatrix} = (0 2 1)\text{.} \end{align*}
Proof.

Let \(G\) be a group. We must find a group of permutations \(\overline{G}\) that is isomorphic to \(G\text{.}\) For any \(g \in G\text{,}\) define a function \(\lambda_g : G \rightarrow G\) by \(\lambda_g(a) = ga\text{.}\) We claim that \(\lambda_g\) is a permutation of \(G\text{.}\) To show that \(\lambda_g\) is one-to-one, suppose that \(\lambda_g(a) = \lambda_g(b)\text{.}\) Then

\begin{equation*} ga =\lambda_g(a) = \lambda_g(b) = gb\text{.} \end{equation*}

Hence, \(a = b\text{.}\) To show that \(\lambda_g\) is onto, we must prove that for each \(a \in G\text{,}\) there is a \(b\) such that \(\lambda_g (b) = a\text{.}\) Let \(b = g^{-1} a\text{.}\)

Now we are ready to define our group \(\overline{G}\text{.}\) Let

\begin{equation*} \overline{G} = \{ \lambda_g : g \in G \}\text{.} \end{equation*}

We must show that \(\overline{G}\) is a group under composition of functions and find an isomorphism between \(G\) and \(\overline{G}\text{.}\) We have closure under composition of functions since

\begin{equation*} (\lambda_g \circ \lambda_h )(a) = \lambda_g(ha) = gha = \lambda_{gh} (a)\text{.} \end{equation*}

Also,

\begin{equation*} \lambda_e (a) = ea = a \end{equation*}

and

\begin{equation*} (\lambda_{g^{-1}} \circ \lambda_g) (a) = \lambda_{g^{-1}} (ga) = g^{-1} g a = a = \lambda_e (a)\text{.} \end{equation*}

We can define an isomorphism from \(G\) to \(\overline{G}\) by \(\phi : g \mapsto \lambda_g\text{.}\) The group operation is preserved since

\begin{equation*} \phi(gh) = \lambda_{gh} = \lambda_g \lambda_h = \phi(g) \phi(h)\text{.} \end{equation*}

It is also one-to-one, because if \(\phi(g)(a) = \phi(h)(a)\text{,}\) then

\begin{equation*} ga = \lambda_g a = \lambda_h a= ha\text{.} \end{equation*}

Hence, \(g = h\text{.}\) That \(\phi\) is onto follows from the fact that \(\phi( g ) = \lambda_g\) for any \(\lambda_g \in \overline{G}\text{.}\)

The isomorphism \(g \mapsto \lambda_g\) is known as the left regular representation of \(G\text{.}\)

Subsection 4.1.2 Historical Note

Arthur Cayley was born in England in 1821, though he spent much of the first part of his life in Russia, where his father was a merchant. Cayley was educated at Cambridge, where he took the first Smith's Prize in mathematics. A lawyer for much of his adult life, he wrote several papers in his early twenties before entering the legal profession at the age of 25. While practicing law he continued his mathematical research, writing more than 300 papers during this period of his life. These included some of his best work. In 1863 he left law to become a professor at Cambridge. Cayley wrote more than 900 papers in fields such as group theory, geometry, and linear algebra. His legal knowledge was very valuable to Cambridge; he participated in the writing of many of the university's statutes. Cayley was also one of the people responsible for the admission of women to Cambridge.

Reading Questions 4.1.3 Reading Questions

1.

List three properties of a group that are preserved by isomorphism.

2.

True or false: for two groups to be isomorphic, they must have the same operation. Briefly explain.

3.

How many cyclic groups of order 10 are there? Briefly explain.

4.

After reading the section, what questions do you still have? Write at least one well formulated question (even if you think you understand everything).

Exercises 4.1.4 Practice Problems

1.

Prove that \(\Z \equiv 2\Z\text{.}\)

2.

I'm thinking of a group \(G = \{a, b, c, d, e, f, g, h\}\) with an operation I'm calling \(\oplus\text{.}\) It turns out that my group is isomorphic to \(\Z_8\text{,}\) and that there is an isomorphism \(f: G \to \Z_8\text{.}\)

  1. In \(G\text{,}\) \(a \oplus b = c\text{.}\) Also, \(f(a) = 3\) and \(f(b) = 4\text{.}\) What is \(f(c)\text{?}\)

  2. If \(f(d) = 2\text{,}\) what is \(d\oplus d\text{?}\) Further, prove that \(d\inv = a\oplus a\text{.}\)

3.

Show that \(U(5)\) is isomorphic to \(U(10)\text{,}\) but \(U(12)\) is not.

4.

Let \(\phi:G \to H\) be an isomorphism. Show that \(\phi(x) = e_H\) if and only if \(x = e_G\text{,}\) where \(e_G\) and \(e_H\) are the identities of \(G\) and \(H\text{,}\) respectively.

Hint.

Since \(\phi\) is a bijection, it is enough to prove that \(\phi(e_G) = e_H\text{.}\) Note that \(e_G = e_G^2\text{.}\) Use the homomorphism property.

5.

Let \(G = \R \setminus \{-1\}\) with the operation \(\ast\) defined by

\begin{equation*} a \ast b = a+b+ab. \end{equation*}

We previously saw that \(G\) with this operation is a group. Show that \((G, \ast)\) is isomprhic to the multiplicative group of nonzero real numbers \(\R^*\text{.}\)

Hint.

You need to define the isomorphism \(f:G \to \R^*\text{.}\) A good place to start would be to decide what the identity is in each group. Guess a reasonable function that sends the identity in \(G\) to the identity in \(\R^*\text{.}\)

6.

True or false: if \(\phi:G \to H\) is an isomorphism, then \(\phi(a^n) = (\phi(a))^n\) for all \(a \in G\text{.}\)

Exercises 4.1.5 Collected Homework

C1.

Consider the groups \(\Z_6\) and \(\Z_2\times \Z_3\text{.}\)

  1. Is there an isomorphism \(f: \Z_6 \to \Z_2\times \Z_3\) that has \(f(1) = (0,1)\text{?}\) Explain why or why not.

  2. Determine whether these two groups are isomorphic. Either find an isomorphism (and explain why it is) or explain why they are not isomorphic.

Hint.

First, \(\Z_2\times \Z_3 = \{(0,0), (0,1), (0,2), (1,0), (1,1), (1,2)\}\text{,}\) and the operation is addition coordinate-wise. So \((1,1) + (1,1) = (0,2)\text{.}\)

For the first part, what would \(f(2)\) and \(f(3)\) be if we defined \(f(1) = (0,1)\text{?}\) Why is this a problem?

C2.

Suppose group \(G_1\) and \(G_2\) are isomorphic, with isomorphism \(\phi:G_1 \to G_2\text{.}\) Let \(H_1\) be a subgroup of \(G_1\text{.}\) Define \(H_2 = \phi(H_1) = \{x \in G_2 \st x = \phi(h) \text{ for some } h \in H_1\}\text{.}\) Prove that \(H_2\) is a subgroup of \(G_2\text{.}\) Futher, prove that if \(H_1\) is normal in \(G_1\text{,}\) then \(H_2\) is normal in \(G_2\)

Hint.

This should be a fairly standard subgroup proof. That is, identity, closures, inverses! For the normal part, show \(H_2\) is closed under conjugates (assuming \(H_1\) is).