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(Law of Exponents for Abelian Groups) Let a and b be

Chapter 2, Problem 23E

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QUESTION:

(Law of Exponents for Abelian Groups) Let a and b be elements of an Abelian group and let n be any integer. Show that (ab)n = anbn. Is this also true for non-Abelian groups?

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QUESTION:

(Law of Exponents for Abelian Groups) Let a and b be elements of an Abelian group and let n be any integer. Show that (ab)n = anbn. Is this also true for non-Abelian groups?

ANSWER:

Step 1 of 4

Let  be an abelian group and  are two elements of .

The Mathematical induction can be used to prove .

For ,

                     

Here,  is identity elements of .

For ,

                     

Therefore, the statement is true of .

 

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