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t Z(p®) is an infinite gr

t Z(p®) is an infinite gr

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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please answer the the last item, the encircle red.
1. Let p e Z be a prime number and set Zp = {=E Q: If gcd(n, m) = 1, then p {m}. a. Show that Z, 4Q. b. Let Z(p) be the quotient group Q/Zp. Show that Z(p) is an infinite group. Hint: Show that Z(p®) contains an infinite subset.) c. Recall that a p-group is a group whose elements all have order some power of p. Show that the quotient group Z(p®) is a p-group. d. Show that every proper subgroup of Z(p) is finite. Hint: Show that if U < Z(p®) is infinite, then U = Z(p®). 1 e. Let n e Z+ and Un = + Zp Show that any subgroup W of Z(p®) of finite order is of the form Un pn for some n E Z+. Hint: choose an element in W whose order is maximal (why is this possible?) and use Lagrange's Theorem. d. Show that Uo = {e} C U1 C U2 ….…C Un C..C Z(p®). Conclude from 1d and le that the Un's are the only proper subgroups of Z(p®).
Transcribed Image Text:1. Let p e Z be a prime number and set Zp = {=E Q: If gcd(n, m) = 1, then p {m}. a. Show that Z, 4Q. b. Let Z(p) be the quotient group Q/Zp. Show that Z(p) is an infinite group. Hint: Show that Z(p®) contains an infinite subset.) c. Recall that a p-group is a group whose elements all have order some power of p. Show that the quotient group Z(p®) is a p-group. d. Show that every proper subgroup of Z(p) is finite. Hint: Show that if U < Z(p®) is infinite, then U = Z(p®). 1 e. Let n e Z+ and Un = + Zp Show that any subgroup W of Z(p®) of finite order is of the form Un pn for some n E Z+. Hint: choose an element in W whose order is maximal (why is this possible?) and use Lagrange's Theorem. d. Show that Uo = {e} C U1 C U2 ….…C Un C..C Z(p®). Conclude from 1d and le that the Un's are the only proper subgroups of Z(p®).
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