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2.5. COSETS AND LAGRANGE’S THEOREM
121
This computation requires a little labor. If you want, you can get a
computer to do some of the repetitive work; for example, programs for
computations in the symmetric group are distributed with the symbolic
mathematics program
Mathematica
.
With the notation
H
D f
±
2
S
4
W
±.4/
D
4
g
, the results are
eH
D
.1 2/H
D
.1 3/H
D
.2 3/H
D
.1 2 3/H
D
.1 3 2/H
D
H
.4 3/H
D
.4 3 2/H
D
.2 1/.4 3/H
D
.2 4 3 1/H
D
.4 3 2 1/H
D
.4 3 1/H
D f
.4 3/; .4 3 2/; .2 1/.4 3/; .2 4 3 1/; .4 3 2 1/; .4 3 1/
g
.4 2/H
D
.3 4 2/H
D
.4 2 1/H
D
.4 2 3 1/H
D
.3 4 2 1/H
D
.3 1/.4 2/H
D f
.4 2/; .3 4 2/; .4 2 1/; .4 2 3 1/; .3 4 2 1/; .3 1/.4 2/
g
.4 1/H
D
.4 1/ .3 2/H
D
.2 4 1/H
D
.2 3 4 1/H
D
.3 2 4 1/H
D
.3 4 1/H
D f
.4 1/; .4 1/ .3 2/; .2 4 1/; .2 3 4 1/; .3 2 4 1/; .3 4 1/
g
:
The regularity of the preceding data for left cosets of subgroups of
symmetric groups is striking! Based on these data, can you make any
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
 Fall '08
 EVERAGE
 Algebra, Sets

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