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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! 1