What is the order of the group ℤ10 x ℤ15 x U(10) /〈(9, 10, 7)〉?
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What is the order of the group ℤ10 x ℤ15 x U(10) /〈(9, 10, 7)〉?
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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?Prove that Ca=Ca1, where Ca is the centralizer of a in the group G.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.
- 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 .Exercises 8. Find an isomorphism from the group in Example of this section to the multiplicative group . Sec. 16. Prove that each of the following sets is a subgroup of , the general linear group of order over .Exercises 31. Let be a group with its center: . Prove that if is the only element of order in , then .