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Suppose that p is the smallest prime that divides |G|. Show that any
subgroup of index p in G is normal in G.
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- 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.Suppose that G is a finite group. Prove that each element of G appears in the multiplication table for G exactly once in each row and exactly once in each column.10. Let be an integer, and let be a fixed integer. Prove or disprove that the set, is subgroup of under addition.
- 15. Prove that on a given collection of groups, the relation of being a homomorphic image has the reflexive property.31. Prove statement of Theorem : for all integers and .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.