Question

Give an example of elements a and b from a group such that a has finite order, b has infinite order and ab has finite order.

Answer 1

An example of elements a and b from a group such that a has finite order, b has infinite order, and ab has finite order is provided using the group of integers under addition.

Step-by-step explanation:

An example of elements a and b from a group such that a has finite order, b has infinite order, and ab has finite order is:

1. 
   
3. Consider the group of integers under addition. Let a = 2, which has a finite order of 1 since 2 + 2 = 4, and then 4 + 2 = 6, and so on.
4. 
   
6. Let b = 3, which has an infinite order since 3 + 3 = 6, and then 6 + 3 = 9, and so on.
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9. Now, compute ab: 2 + 3 = 5, which has a finite order of 1 since 5 + 2 = 7, and then 7 + 2 = 9, and so on.