If the order of a group G is p²q² with p and q as distinct primes, then G is not simple.
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- If a is an element of order m in a group G and ak=e, prove that m divides k.Use mathematical induction to prove that if a is an element of a group G, then (a1)n=(an)1 for every positive integer n.Use mathematical induction to prove that if a1,a2,...,an are elements of a group G, then (a1a2...an)1=an1an11...a21a11. (This is the general form of the reverse order law for inverses.)
- Let n be appositive integer, n1. Prove by induction that the set of transpositions (1,2),(1,3),...,(1,n) generates the entire group Sn.19. a. Show that is isomorphic to , where the group operation in each of , and is addition. b. Show that is isomorphic to , where all group operations are addition.Find the order of each of the following elements in the multiplicative group of units . for for for for
- 9. Find all homomorphic images of the octic group.20. Let and be elements of a group . Use mathematical induction to prove each of the following statements for all positive integers . a. If the operation is multiplication, then . b. If the operation is addition and is abelian , then .Exercises 30. For an arbitrary positive integer, prove that any two cyclic groups of order are isomorphic.
- Exercises 11. According to Exercise of section, if is prime, the nonzero elements of form a group with respect to multiplication. For each of the following values of , show that this group is cyclic. (Sec. ) a. b. c. d. e. f. 33. a. Let . Show that is a group with respect to multiplication in if and only if is a prime. State the order of . This group is called the group of units in and designated by . b. Construct a multiplication table for the group of all nonzero elements in , and identify the inverse of each element.If p1,p2,...,pr are distinct primes, prove that any two abelian groups that have order n=p1p2...pr are isomorphic.Prove that any group with prime order is cyclic.