Prove that if n is not prime, then {1, 2, 3,..., n-1} is not a group under multiplication mod n.
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Prove that if n is not prime, then {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.Prove that any group with prime order is cyclic.
- If a is an element of order m in a group G and ak=e, prove that m divides k.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 .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 .
- 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 a subset of Z that is closed under addition but is not subgroup of the additive group Z.Find all Sylow 3-subgroups of the symmetric group S4.
- Prove or disprove that the set of all diagonal matrices in Mn() forms a group with respect to addition.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.)Exercises In Exercises, decide whether each of the given sets is a group with respect to the indicated operation. If it is not a group, state a condition in Definition that fails to hold. 8. For a fixed positive integer, the set of all complex numbers such that (that is, the set of all roots of), with operation multiplication.