(b) Let G be a group and let x E G. Suppose that (x) is a cyclic group of order N. Let m and n be positive integers. We know that (x) and (x") are subgroups of (x), hence their intersection (rm) n (r) is a subgroup of (r). As proved in class, any subgroup of a cyclic group is cyclic, so (rm) n (x) is cyclic. Find a generator. (It will be xP for some choice of p. Your job is to find p.) Justify your answer. (It may help to work through some
(b) Let G be a group and let x E G. Suppose that (x) is a cyclic group of order N. Let m and n be positive integers. We know that (x) and (x") are subgroups of (x), hence their intersection (rm) n (r) is a subgroup of (r). As proved in class, any subgroup of a cyclic group is cyclic, so (rm) n (x) is cyclic. Find a generator. (It will be xP for some choice of p. Your job is to find p.) Justify your answer. (It may help to work through some
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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Question
I need help with 6b
Transcribed Image Text:6.
(a) Let G be a group and let x E G. Suppose that (x) is an infinite cyclic group. Let
m and n be positive integers. We know that (x) and (x") are subgroups of (x), hence their
intersection (m) n (rn) is a subgroup of (x). As proved in class, any subgroup of a cyclic
group is cyclic, so (rm) n(x) is cyclic. Find a generator. (It will be x for some choice of
p. Your job is to find p.) Justify your answer. (It may help to work through some examples
as a warm-up.)
(b) Let G be a group and let x E G. Suppose that (x) is a cyclic group of order N.
Let m and n be positive integers. We know that (x) and (x") are subgroups of (x), hence
their intersection (rm) n(x) is a subgroup of (x). As proved in class, any subgroup of a
cyclic group is cyclic, so (xm) n(x) is cyclic. Find a generator. (It will be xP for some
choice of p. Your job is to find p.) Justify your answer. (It may help to work through some
examples as a warm-up. Note that we are not assuming that m and n are divisors of N.)
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