Problem 3. Let G be a group. Let Z(G) denote the set Z(G) = {z E G: Vg E G, zg = gz} %3D is a subgroup of G. Give an example of a group where Z(G) # G. The subset Z(G) is called the center of the group G.
Problem 3. Let G be a group. Let Z(G) denote the set Z(G) = {z E G: Vg E G, zg = gz} %3D is a subgroup of G. Give an example of a group where Z(G) # G. The subset Z(G) is called the center of the group G.
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
![sheet-01-25.pdf
sheet-01-25.pdf (127 KB)
Page <
of 2
Solution.
(1) Using the given relations we can directly compute
Ho²up = µp²(Hp) = µ0°pµ= (µp)µ = p='4? = p=1 = p°.
(2) First note that we have
(HP) o p² = µp3
på o (HP) = p*µp = ppp= µp°p=µp".
If p and up commute then the above computation tells us that
(Hp) o p = pt o (µp) = up° = Hp =p= L.
However p has order 7 in D7 so p=t is not possible. We conclude that the
two given elements do notrcommute.
Problem 3. Let G be a group. Let Z(G) denote the set
Z(G) = {z e G: Vg E G, zg = gz}
is a subgroup of G. Give an example of a group where Z(G)+ G. The subset Z(G)
is called the center of the group G.
Lecture8Notes.pdf
cse11-pa3-starte..zip
Lecture7Notes.pdf
Aa
MacBook Pro
2.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F67a79aa2-f715-406c-8bee-8178252bb86d%2F3b05ee1b-599b-49a2-adb1-165c5101d28b%2Foyptpd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:sheet-01-25.pdf
sheet-01-25.pdf (127 KB)
Page <
of 2
Solution.
(1) Using the given relations we can directly compute
Ho²up = µp²(Hp) = µ0°pµ= (µp)µ = p='4? = p=1 = p°.
(2) First note that we have
(HP) o p² = µp3
på o (HP) = p*µp = ppp= µp°p=µp".
If p and up commute then the above computation tells us that
(Hp) o p = pt o (µp) = up° = Hp =p= L.
However p has order 7 in D7 so p=t is not possible. We conclude that the
two given elements do notrcommute.
Problem 3. Let G be a group. Let Z(G) denote the set
Z(G) = {z e G: Vg E G, zg = gz}
is a subgroup of G. Give an example of a group where Z(G)+ G. The subset Z(G)
is called the center of the group G.
Lecture8Notes.pdf
cse11-pa3-starte..zip
Lecture7Notes.pdf
Aa
MacBook Pro
2.
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