All of the following groups have order 4. Gather the groups that are isomorphic to ea other in separate collections. (In other words, partition the groups into isomorphis classes.) is0 to 7 Z4 Cyclic v Z2 x Z2 are any o dudes {0, 2, 4, 6} in Z8 {ro, r2, d1, d2} in D4 {1, –1, i, –i} in C Us U10 U12 {1, 4, 11, 14} in U15 {1,3, 9, 11} in U16 {0, 1, x, x+ 1} in Z2[x] (polynomial addition over Zz) in (matrix addition, binary entries) 0 21 in GL(2, Z3) (multiplication, entries in Z3) L0 2] (See similar example on page 179.)
All of the following groups have order 4. Gather the groups that are isomorphic to ea other in separate collections. (In other words, partition the groups into isomorphis classes.) is0 to 7 Z4 Cyclic v Z2 x Z2 are any o dudes {0, 2, 4, 6} in Z8 {ro, r2, d1, d2} in D4 {1, –1, i, –i} in C Us U10 U12 {1, 4, 11, 14} in U15 {1,3, 9, 11} in U16 {0, 1, x, x+ 1} in Z2[x] (polynomial addition over Zz) in (matrix addition, binary entries) 0 21 in GL(2, Z3) (multiplication, entries in Z3) L0 2] (See similar example on page 179.)
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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![All of the following groups have order 4. Gather the groups that are isomorphic to each
other in separate collections. (In other words, partition the groups into isomorphism
classes.)
is0
to Lu
Z4
Cyclic us r
Z, x Z2
dudes e
{0, 2, 4, 6} in Z8
{ro, r2, d1, d2} in D4
{1, –1, i, -i} in ¢
Us
U10
U12
{1,4, 11, 14} in U15
{1,3,9, 11} in U16
{0, 1, x, x + 1} in Z2[x] (polynomial addition over Z2)
{1: 1-6 1:18 1-6 } in Ma(Z,) (matrix addition, binary entries)
[O 1
[2 01 [0 21
{l. 1:R LE 2 ) in GL(2, Z.) (multiplication, entries in Z3)
(See similar example on page 179.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe9dbf5ef-3980-409f-876a-bbf2fd708a69%2Faee36f60-155b-4f2b-bc82-c6ebd9b66501%2Fpeibx0n_processed.jpeg&w=3840&q=75)
Transcribed Image Text:All of the following groups have order 4. Gather the groups that are isomorphic to each
other in separate collections. (In other words, partition the groups into isomorphism
classes.)
is0
to Lu
Z4
Cyclic us r
Z, x Z2
dudes e
{0, 2, 4, 6} in Z8
{ro, r2, d1, d2} in D4
{1, –1, i, -i} in ¢
Us
U10
U12
{1,4, 11, 14} in U15
{1,3,9, 11} in U16
{0, 1, x, x + 1} in Z2[x] (polynomial addition over Z2)
{1: 1-6 1:18 1-6 } in Ma(Z,) (matrix addition, binary entries)
[O 1
[2 01 [0 21
{l. 1:R LE 2 ) in GL(2, Z.) (multiplication, entries in Z3)
(See similar example on page 179.)
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