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Math 404 Chapter 9 Homework Due October 31 You may need to use the facts from Exercise B in the text. (Make sure you are doing the practice problems, by the way!) 1. Suppose that G ? H. Show that if G is abelian, then so is H. 2. Suppose that f : G ? H is an isomorphism. Show that f(a^n) = [f(a)]^n for all n ? N. You will need to use induction. 3. Suppose that G is the set of non-zero real numbers with the operation x * y = 3xy. Show that G ? ?*. Hint: G is a group; in G the identity element is 1/3 and the inverse of x is 1/9x you might want to convince yourself of these facts before attempting the proof. 4. Consider the subgroup H of ? given by G = {a + b?2 : a, b ? ?} (you proved that this is a group in your homework from Chapter 5). Show that H ? ? × ?. 5. Show that the function f : S4 ? S4 given by f(?) = (123)?(132) is an isomorphism from S4 onto itself. To clarify the definition (maybe), here's an example: f(134) = (123)(134)(132) = (142).

          Math 404
Chapter 9 Homework
Due October 31
You may need to use the facts from Exercise B in the text. (Make sure you are doing the practice problems, by the way!)
1. Suppose that G ? H. Show that if G is abelian, then so is H.
2. Suppose that f : G ? H is an isomorphism. Show that f(a^n) = [f(a)]^n for all n ? N.
You will need to use induction.
3. Suppose that G is the set of non-zero real numbers with the operation x * y = 3xy. Show that G ? ?*. Hint: G is a group; in G the identity element is 1/3 and the inverse of x is 1/9x you might want to convince yourself of these facts before attempting the proof.
4. Consider the subgroup H of ? given by G = {a + b?2 : a, b ? ?} (you proved that this is a group in your homework from Chapter 5). Show that H ? ? × ?.
5. Show that the function f : S4 ? S4 given by f(?) = (123)?(132) is an isomorphism from S4 onto itself. To clarify the definition (maybe), here's an example:
f(134) = (123)(134)(132) = (142).

Math 404
Chapter 9 Homework
Due October 31
You may need to use the facts from Exercise B in the text. (Make sure you are doing the practice problems, by the way!)
1. Suppose that G ? H. Show that if G is abelian, then so is H.
2. Suppose that f : G ? H is an isomorphism. Show that f(a^n) = [f(a)]^n for all n ? N.
You will need to use induction.
3. Suppose that G is the set of non-zero real numbers with the operation x * y = 3xy. Show that G ? ?*. Hint: G is a group; in G the identity element is 1/3 and the inverse of x is 1/9x you might want to convince yourself of these facts before attempting the proof.
4. Consider the subgroup H of ? given by G = a + b?2 : a, b ? ? (you proved that this is a group in your homework from Chapter 5). Show that H ? ? × ?.
5. Show that the function f : S4 ? S4 given by f(?) = (123)?(132) is an isomorphism from S4 onto itself. To clarify the definition (maybe), here's an example:
f(134) = (123)(134)(132) = (142).

Added by Esperanza C.

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