Problem 1.0.3. Let a e G then we define the cyclic subgroup generated by a to be < a >:= {a"|n E Z} Some comments regarding the definition: aº = e where e is the identity element of the group. For n > 0, a" = a ★ a * · ·. * a (n-times) where * is the multiplication/operation of the group. ... Also note that n E Z so a-1 e< a > and for n < 0 we define a" (|n|-times). So for example a-3 = a-1 ★ a-1 ★a-1. (a-1) * ·.. ★ a-1 a) Let (G, *) = (Z,+) (integers with respect to addition) describe the elements of <1>, what is < 3 >? b) Let (G, *) = (Zs,+s) (residue classes modulo 8 with respect to the addition modulo 8). Find all such [a] such that < [a] >= Zs. This set of [a]'s should look familiar from somehwere..
Problem 1.0.3. Let a e G then we define the cyclic subgroup generated by a to be < a >:= {a"|n E Z} Some comments regarding the definition: aº = e where e is the identity element of the group. For n > 0, a" = a ★ a * · ·. * a (n-times) where * is the multiplication/operation of the group. ... Also note that n E Z so a-1 e< a > and for n < 0 we define a" (|n|-times). So for example a-3 = a-1 ★ a-1 ★a-1. (a-1) * ·.. ★ a-1 a) Let (G, *) = (Z,+) (integers with respect to addition) describe the elements of <1>, what is < 3 >? b) Let (G, *) = (Zs,+s) (residue classes modulo 8 with respect to the addition modulo 8). Find all such [a] such that < [a] >= Zs. This set of [a]'s should look familiar from somehwere..
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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![Problem 1.0.3. Let a e G then we define the cyclic subgroup generated by
a to be
< a >:= {a"|n E Z}
Some comments regarding the definition: aº = e where e is the identity element of
the group.
For n > 0, a" = a ★ a * · ·. * a (n-times) where * is the multiplication/operation of
the group.
...
Also note that n E Z so a-1 e< a > and for n < 0 we define a"
(|n|-times). So for example a-3 = a-1 ★ a-1 ★a-1.
(a-1) * ·.. ★ a-1
a) Let (G, *) = (Z,+) (integers with respect to addition) describe the elements of
<1>, what is < 3 >?
b) Let (G, *) = (Zs,+s) (residue classes modulo 8 with respect to the addition
modulo 8). Find all such [a] such that
< [a] >= Zs. This set of [a]'s should look familiar from somehwere..](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffc8433cf-bcdb-414d-b469-0cfa6525aac7%2F7bb9c828-1f20-46b8-ac24-fc946c3c51a7%2F682j1oc_processed.png&w=3840&q=75)
Transcribed Image Text:Problem 1.0.3. Let a e G then we define the cyclic subgroup generated by
a to be
< a >:= {a"|n E Z}
Some comments regarding the definition: aº = e where e is the identity element of
the group.
For n > 0, a" = a ★ a * · ·. * a (n-times) where * is the multiplication/operation of
the group.
...
Also note that n E Z so a-1 e< a > and for n < 0 we define a"
(|n|-times). So for example a-3 = a-1 ★ a-1 ★a-1.
(a-1) * ·.. ★ a-1
a) Let (G, *) = (Z,+) (integers with respect to addition) describe the elements of
<1>, what is < 3 >?
b) Let (G, *) = (Zs,+s) (residue classes modulo 8 with respect to the addition
modulo 8). Find all such [a] such that
< [a] >= Zs. This set of [a]'s should look familiar from somehwere..
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