If a group G has a composition series then any two composition series of G are isomorphic.
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- 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.27. a. Show that a cyclic group of order has a cyclic group of order as a homomorphic image. b. Show that a cyclic group of order has a cyclic group of order as a homomorphic image.9. Find all homomorphic images of the octic group.
- 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 .True or False Label each of the following statements as either true or false. 6. Every permutation can be expressed as a product of disjoint cycles.True or False Label each of the following statements as either true or false. In a Cayley table for a group, each element appears exactly once in each row.
- 11. Find all normal subgroups of the alternating group .12. Find all homomorphic images of each group in Exercise of Section. 18. Let be the group of units as described in Exercise. For each value of, write out the elements of and construct a multiplication table for . a. b. c. d.Exercises 19. Find cyclic subgroups of that have three different orders.