If n is not prime, then G = {1, 2, 3,..., n-1} is not a group under multiplication mod n.
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If n is not prime, then G = {1, 2, 3,..., n-1} is not a group under multiplication mod n.
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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.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.If a is an element of order m in a group G and ak=e, prove that m divides k.
- Find all Sylow 3-subgroups of the symmetric group S4.Prove that any group with prime order is cyclic.15. Assume that can be written as the direct sum , where is a cyclic group of order . Prove that has elements of order but no elements of order greater than Find the number of distinct elements of that have order .
- Let H and K be subgroups of a group G and K a subgroup of H. If the order of G is 24 and the order of K is 3, what are all the possible orders of H?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 .Suppose that the abelian group G can be written as the direct sum G=C22C3C3, where Cn is a cyclic group of order n. Prove that G has elements of order 12 but no element of order greater than 12. Find the number of distinct elements of G that have order 12.