9.1: Definition and Examples
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Two groups (G,⋅) and (H,∘) are isomorphic if there exists a one-to-one and onto map ϕ:G→H such that the group operation is preserved; that is,
for all a and b in G. If G is isomorphic to H, we write G≅H. The map ϕ is called an isomorphism.
To show that Z4≅⟨i⟩, define a map ϕ:Z4→⟨i⟩ by ϕ(n)=in. We must show that ϕ is bijective and preserves the group operation.
Solution
The map ϕ is one-to-one and onto because
Since
the group operation is preserved.
We can define an isomorphism ϕ from the additive group of real numbers (R,+) to the multiplicative group of positive real numbers (R+,⋅) with the exponential map;
Solution
that is,
Of course, we must still show that ϕ is one-to-one and onto, but this can be determined using calculus.
The integers are isomorphic to the subgroup of Q∗ consisting of elements of the form 2n. Define a map ϕ:Z→Q∗ by ϕ(n)=2n.
Solution
Then
By definition the map ϕ is onto the subset {2n:n∈Z} of Q∗. To show that the map is injective, assume that m≠n. If we can show that ϕ(m)≠ϕ(n), then we are done. Suppose that m>n and assume that ϕ(m)=ϕ(n). Then 2m=2n or 2m−n=1, which is impossible since m−n>0.
The groups Z8 and Z12 cannot be isomorphic since they have different orders; however, it is true that U(8)≅U(12). We know that
Solution
An isomorphism ϕ:U(8)→U(12) is then given by
The map ϕ is not the only possible isomorphism between these two groups. We could define another isomorphism ψ by ψ(1)=1, ψ(3)=11, ψ(5)=5, ψ(7)=7. In fact, both of these groups are isomorphic to Z2×Z2 (see Example 3.28 in Chapter 3).
Even though S3 and Z6 possess the same number of elements, we would suspect that they are not isomorphic, because Z6 is abelian and S3 is nonabelian. To demonstrate that this is indeed the case, suppose that ϕ:Z6→S3 is an isomorphism.
Solution
Let a,b∈S3 be two elements such that ab≠ba. Since ϕ is an isomorphism, there exist elements m and n in Z6 such that
However,
which contradicts the fact that a and b do not commute.
Let ϕ:G→H be an isomorphism of two groups. Then the following statements are true.
- ϕ−1:H→G is an isomorphism.
- |G|=|H|.
- If G is abelian, then H is abelian.
- If G is cyclic, then H is cyclic.
- If G has a subgroup of order n, then H has a subgroup of order n.
- Proof
-
Assertions (1) and (2) follow from the fact that ϕ is a bijection. We will prove (3) here and leave the remainder of the theorem to be proved in the exercises.
(3) Suppose that h1 and h2 are elements of H. Since ϕ is onto, there exist elements g1,g2∈G such that ϕ(g1)=h1 and ϕ(g2)=h2. Therefore,
h1h2=ϕ(g1)ϕ(g2)=ϕ(g1g2)=ϕ(g2g1)=ϕ(g2)ϕ(g1)=h2h1.
We are now in a position to characterize all cyclic groups.
All cyclic groups of infinite order are isomorphic to Z.
- Proof
-
Let G be a cyclic group with infinite order and suppose that a is a generator of G. Define a map ϕ:Z→G by ϕ:n↦an. Then
ϕ(m+n)=am+n=aman=ϕ(m)ϕ(n).To show that ϕ is injective, suppose that m and n are two elements in Z, where m≠n. We can assume that m>n. We must show that am≠an. Let us suppose the contrary; that is, am=an. In this case am−n=e, where m−n>0, which contradicts the fact that a has infinite order. Our map is onto since any element in G can be written as an for some integer n and ϕ(n)=an.
If G is a cyclic group of order n, then G is isomorphic to Zn.
- Proof
-
Let G be a cyclic group of order n generated by a and define a map ϕ:Zn→G by ϕ:k↦ak, where 0≤k<n. The proof that ϕ is an isomorphism is one of the end-of-chapter exercises.
If G is a group of order p, where p is a prime number, then G is isomorphic to Zp.
- Proof
-
The proof is a direct result of Corollary 6.12.
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.
The isomorphism of groups determines an equivalence relation on the class of all groups
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.
Consider the group Z3. The Cayley table for Z3 is as follows.
The addition table of Z3 suggests that it is the same as the permutation group G={(0),(012),(021)}.
Solution
The isomorphism here is
Every group is isomorphic to a group of permutations.
- Proof
-
Let G be a group. We must find a group of permutations ¯G that is isomorphic to G. For any g∈G, define a function λg:G→G by λg(a)=ga. We claim that λg is a permutation of G. To show that λg is one-to-one, suppose that λg(a)=λg(b). Then
ga=λg(a)=λg(b)=gb.Hence, a=b. To show that λg is onto, we must prove that for each a∈G, there is a b such that λg(b)=a. Let b=g−1a.
Now we are ready to define our group ¯G. Let
¯G={λg:g∈G}.We must show that ¯G is a group under composition of functions and find an isomorphism between G and ¯G. We have closure under composition of functions since
(λg∘λh)(a)=λg(ha)=gha=λgh(a).Also,
λe(a)=ea=aand
(λg−1∘λg)(a)=λg−1(ga)=g−1ga=a=λe(a).We can define an isomorphism from G to ¯G by ϕ:g↦λg. The group operation is preserved since
ϕ(gh)=λgh=λgλh=ϕ(g)ϕ(h).It is also one-to-one, because if ϕ(g)(a)=ϕ(h)(a), then
ga=λga=λha=ha.Hence, g=h. That ϕ is onto follows from the fact that ϕ(g)=λg for any λg∈¯G.
The isomorphism g↦λg is known as the left regular representation of G.
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.