Page 84 - DMTH403_ABSTRACT_ALGEBRA

Unit 6: Group Isomorphism

However, the order of any real number different from ±1 is infinite: and o(1) = 1, o(–1) = 2. Notes
So we reach a contradiction. Therefore, our supposition must be wrong. That is, R* and C* are
not isomorphic.
You must have noticed that the definition of an isomorphism just says that the map is bijective,
i.e., the inverse map exists. It does not tell us any properties of the inverse. The next result does
so.
Theorem 7: If f : G  G is an isomorphism of groups, then f : G  G is also an isomorphism.
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Proof: You know that f is bijective. So, we only need to show that f is a homomorphism. Let a’,
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b’  G and a = f (a’), b = f (b’). Then f(a)= a’ and f(b)= b’.
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Therefore, f(ab) = f(a) f(b) = a’b’. On applying f , we get
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f (a’b’) = ab = f (a’) f (b’), Thus,
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f (a’b’) = f (a’) f (b’) for all a’, b’  G . 2
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Hence, f is an isomorphism.
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From Theorem 7 we can immediately say that
 : T  R :  (f ) = (a, b) is an isomorphism.
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a,b,
Theorem 7 says that if G  G , then G  G . We will be using this result quite often.
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6.3 Group Isomorphism
In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-
one correspondence between the elements of the groups in a way that respects the given group
operations. If there exists an isomorphism between two groups, then the groups are called
isomorphic. From the standpoint of group theory, isomorphic groups have the same properties
and need not be distinguished.
Definition and Notation
Given two groups (G, *) and (H,  ), a group isomorphism from (G, *) to (H,  ) is a bijective
group homomorphism from G to H. Spelled out, this means that a group isomorphism is a
bijective function f : G  H such that for all u and v in G it holds that
f(u * v) = f(u)  f(v).
The two groups (G, *) and (H,  ) are isomorphic if an isomorphism exists. This is written:
(G, *)  (H,  )
Often shorter and more simple notations can be used. Often there is no ambiguity about the
group operation, and it can be omitted:
G  H
Sometimes one can even simply write G = H. Whether such a notation is possible without
confusion or ambiguity depends on context. For example, the equals sign is not very suitable
when the groups are both subgroups of the same group.
Conversely, given a group (G, *), a set H, and a bijection f : G  H, we can make H a group
(H,  ) by defining
f(u)  f(v) = f(u * v).
If H = G and  = * then the bijection is an automorphism (q.v.)

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