Given H = a) Show that His a subgroup of A4- b) Find all the left cosets of H in A4. c) Find all the right cosets of H in A4. d) Does aH=Ha for all a in A4? {(1), (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)}.
Given H = a) Show that His a subgroup of A4- b) Find all the left cosets of H in A4. c) Find all the right cosets of H in A4. d) Does aH=Ha for all a in A4? {(1), (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)}.
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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Transcribed Image Text:TABLE 5.1
The Alternating Group A4 of Even Permutations of {1, 2, 3, 4}
(In this table, the permutations of A4 are designated as 1, a2, . . . , &12 and an entry k
inside the table represents ag. For example, az ag = a6.)
a4
a6
ag
a10
d12
(1) = a1
(12)(34) = a2
(13)(24) = az
(14)(23) = a4
(123) = a5
(243) = a6
1
2
3
4
7
8.
9.
10
11
12
4
6.
2
3
8
7
10
9.
12
3
%3D
7
8.
6
11
12
9.
3
1
8.
6 5 12
7
11
10
8.
6.
9.
12
10
11
1
4
6.
(142) = a7
(134) = ag
(132) = ¤9
7 5 8
10
11
12
2
3
7
6 8 5
11
10
12
9.
3
8
5
6
12
9.
11
10
1
9.
11
12
10
1
4
2
5
7
(143)
10
%3D
a10
12
11
9.
2
4
3
1
6.
8.
(234) = ¤11
11
9.
10
12
9.
(124) = a12
3
1
4
5
6.
12
10
11
4.
1
3
8
S=O9 34126507
O214387 65
1 2
Transcribed Image Text:(13)(24) = a3
(14)(23) = a4
(123) = as
(243) = a6
4
3
2
1
6.
7.5
8.
6.
7.
8.
%3D
(142) = a7
(134) = ag
(132) = a9
(143) = a10
(234) = a11
12
6.
8.
8.
6.
%3D
9 11
12
10
%3D
10
12
11 9
11
6.
10
12
(124) = a12
10
9.
11
%3D
Given H = {(1), (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)}.
a) Show that His a subgroup of A4-
b) Find all the left cosets of H in A4.
c) Find all the right cosets of H in A4-
d) Does aH =Ha for all a in A4?
345
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