Show that * defined on Z by a*b=|a+b| is not a group. (Hint: identify and show the group property that is not satisfied).
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Show that * defined on Z by a*b=|a+b| is not a group. (Hint: identify and show the group property that is not satisfied).
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- If a is an element of order m in a group G and ak=e, prove that m divides k.31. (See Exercise 30.) Prove that if and are primes and is a nonabelian group of order , then the center of is the trivial subgroup . Exercise 30: 30. Let be a group with center . Prove that if is cyclic, then is abelian.Exercises 31. Let be a group with its center: . Prove that if is the only element of order in , then .
- 12. Find all homomorphic images of each group in Exercise of Section. 18. Let be the group of units as described in Exercise. For each value of, write out the elements of and construct a multiplication table for . a. b. c. d.Show that every subgroup of an abelian group is normal.Exercises 18. Suppose and let be defined by . Prove or disprove that is an automorphism of the additive group .