Let G be a group with order p where p is a prime. How many elements has order p?Explain.
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Let G be a group with order p where p is a prime. How many elements has order p?
Explain.
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- 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?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.Let a,b,c, and d be elements of a group G. Find an expression for (abcd)1 in terms of a1,b1,c1, and d1.
- 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.9. Find all homomorphic images of the octic group.Construct a multiplication table for the group D5 of rigid motions of a regular pentagon with vertices 1,2,3,4,5.
- 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 .The alternating group A4 on 4 elements is the same as the group D4 of symmetries for a square. That is. A4=D4.Exercises 21. Suppose is a cyclic group of order. Determine the number of generators of for each value of and list all the distinct generators of . a. b. c. d. e. f.