1. Let p e Z be a prime number and set Z, = {- e Q: If gcd(n, m) = 1, then p Im}. %3D a. Show that Z, 4 Q. 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.
1. Let p e Z be a prime number and set Z, = {- e Q: If gcd(n, m) = 1, then p Im}. %3D a. Show that Z, 4 Q. 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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question

Transcribed Image Text:1. Let p e Z be a prime number and set Z, = e
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 ifUSZ(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
for some n E Zt.
Hint: choose an element in W whose order is maximal (why is this possible?) and use
Lagrange's Theorem.
d. Show that U, = {e} C U1 C U2 C C UnC.C Z(p). Conclude from 1d and le that the Un's are
the only proper subgroups of Z(p).
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