(1) Let G=< a> be a finite cyclic group of order n. (a) Suppose that n = dq where d and q are non-negative integers. Show that is the unique cyclic subgroup of order d in G (b) Show that every subgroup of G is of the form < a> for some q that divides n (c) State the fundamental theorem of cyclic groups. (d) Show that if d divides n then there are exactly o(d) elements of order d in G. (e) Let H = U(35) = {[m] € 33z|gcd(m, d) = 1}. Compute ord(6) and ord(-1) in H. Show that H is not a cyclic group.
(1) Let G=< a> be a finite cyclic group of order n. (a) Suppose that n = dq where d and q are non-negative integers. Show that is the unique cyclic subgroup of order d in G (b) Show that every subgroup of G is of the form < a> for some q that divides n (c) State the fundamental theorem of cyclic groups. (d) Show that if d divides n then there are exactly o(d) elements of order d in G. (e) Let H = U(35) = {[m] € 33z|gcd(m, d) = 1}. Compute ord(6) and ord(-1) in H. Show that H is not a cyclic group.
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 do part d) and e)
![(1) Let G=< a> be a finite cyclic group of order n.
(a) Suppose that n = dq where d and q are non-negative integers.
Show that <aª > is the unique cyclic subgroup of order d in G
(b) Show that every subgroup of G is of the form < a> for some
q that divides n
(c) State the fundamental theorem of cyclic groups.
(d) Show that if d divides n then there are exactly o(d) elements of
order d in G.
(e) Let H = U(35) = {[m] € 3z|gcd(m, d) = 1}. Compute ord(6)
and ord(-1) in H. Show that H is not a cyclic group.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffa923b6f-81dd-482c-8885-6de6bc295751%2Fbccf6e07-1a5b-4810-b502-4a5dfdfff9f4%2Fgxobm3j_processed.png&w=3840&q=75)
Transcribed Image Text:(1) Let G=< a> be a finite cyclic group of order n.
(a) Suppose that n = dq where d and q are non-negative integers.
Show that <aª > is the unique cyclic subgroup of order d in G
(b) Show that every subgroup of G is of the form < a> for some
q that divides n
(c) State the fundamental theorem of cyclic groups.
(d) Show that if d divides n then there are exactly o(d) elements of
order d in G.
(e) Let H = U(35) = {[m] € 3z|gcd(m, d) = 1}. Compute ord(6)
and ord(-1) in H. Show that H is not a cyclic group.
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