The order of the element (2,9) in Z₁ × U 10² (Z is the additive group modulo 4 and U₁is Euler group) 10 18 4 9 2
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- 9. Find all homomorphic images of the octic group.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.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 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.)Find the order of each of the following elements in the multiplicative group of units . for for for forFind all subgroups of the octic group D4.
- The alternating group A4 on 4 elements is the same as the group D4 of symmetries for a square. That is. A4=D4.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.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 .