Math 281
Homework
Oct. 31st
1. Complex numbers form a group with the operation of addition. Similarly, R × R is a group, where addition is defined by coordinatewise
addition:
(x1 , y1 ) + (x2 , y2 ) := (x1 + y1 , x2 + y2 )
Show that (C, +) and (R × R, +) are isomorphic groups.
2. Give one example of isomorphism between groups.
3. Can you have an isomorphism between a finite and an infinite group?
Why?
4. Consider the cycle g = g −1 = (13) Compute the following:
• (13)(123456789) (13)
• (13)(23)(13)
• (13)(12453) (13)
• (13)(2345) (13)
• (13)(345) (13)
• (13)(45) (13)
5. Consider the cycle g = (123).
Compute the following:
Now the inverse is g −1 = (132).
• (132)(123456789) (123)
• (132)(23)(123)
• (132)(12453) (123)
• (132)(2345) (123)
• (132)(345) (123)
• (132)(45) (123)
6. Make a (educated) guess about how does conjugation work in the
symmetryc group. If the previous two exercises are not enough for
you to spot a pattern, you might want to do more examples!
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