Let p e Z be a prime number and set Z, = {" e E Q: If ged(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®). 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.
Let p e Z be a prime number and set Z, = {" e E Q: If ged(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®). 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.
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 d and e
Transcribed Image Text:1. Let p e Z be a prime number and set Z, = {-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®).
e. Let n E Z+ and Un =
pn
+ Z,
Show that any subgroup W of Z(p®) of finite order is of the form Un
for some n E Z+.
Hint: choose an element in W whose order is maximal (why is this possible?) and use
Lagrange's Theorem.
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