Elementary Abstract Algebra- Examples and Applications, 2019a

702 CHAPTER 17 ISOMORPHISMS OF GROUPS
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It’s possible to use isomorphisms to create other isomorphisms:
Exercise 17.2.9.
(a) Given that φ : G → H is an isomorphism, show that φ −1 : H → G is
also an isomorphism. (*Hint*)
(b) Given that φ : G → H and ψ : H → K are isomorphisms, show that
ψ ◦ φ : G → K is also an isomorphism. (*Hint*)
We said in the previous section that isomorphic groups are “equivalent”
in some sense. This fact has a formal mathematical statement as well:
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Proposition 17.2.10. Isomorphism is an equivalence relation on groups.
Exercise 17.2.11. Prove Proposition 17.2.10. (*Hint*)
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17.3 Examples and generalizations
17.3.1 Examples of isomorphisms
Now that we have a formal definition of what it means for two groups to be
isomorphic, let’s look at some more examples, in order to get a good feel for
identifying groups that are isomorphic and those that aren’t.
From high school and college algebra we are well familiar with the fact
that when you multiply exponentials (with the same bases), the result of this
operation is the same as if you had just kept the base and added the exponents.
This equivalence of operations is a telltale sign for identifying possible
isomorphic groups. The next two examples illustrate this observation.
For our first example, we denote the set of integer powers of 2 as 2 Z ,
that is:
2 Z ≡{...,2 −2 , 2 −1 , 2 0 , 2 1 , 2 2 ,...}.
Exercise 17.3.1. Show that 2 Z with the operation of multiplication is a