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Find the right & left cosets

  • Thread starter hsong9
  • Start date
80
1
1. Homework Statement
In each case find the right and left cosets in G of the subgroups H and K of G.
a) G = A4; H = {e, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)}, K = <(1 2 3)>
b) G= Z12; H = 3Z12, K = 2Z12
c) G = D4 = D3 = {1,a,a2
, a3, b, ba, ba2, ba3}, |a|=4,|b|=2, and aba = b; H =<a2>, K=<b>
d) G=any group, H is any subgroup of index 2

2. Homework Equations
A4 is alternating group degree 4


3. The Attempt at a Solution

a) H(1 2 3) = {(1 2 3),(2 4 3),(1 4 2),(1 3 4)} = (1 2 3)H
H(1 3 2) = {(1 3 2),(1 4 3),(2 3 4),(1 2 4)} = (1 3 2)H
I know this is correct, but I don't know why
H(1 2 4) = {(1 2 4),(2 3 4),(1 4 3),(1 3 2)}...
H(1 4 2) = {....} are not cosets of G.

b) H + 0 = {3k | k in Z12 }
H + 1 = {3k+1| k in Z12 }
H + 2 = {3k+2 | k in Z12 }
correct??

c) Ha = {1,a,a2, a3, b, ba, ba2, ba3}
aH = {{1,a,a2, a3, b, ab, a2b, a3b}
K1 = {a, ab, a2b, a3b, b, ba2b, ba3b}
1K = {{1,a,a2, a3, b, ba, ba2, ba3}
Correct?

d) This is just Ha and aH ?? I'm not sure about this quesiton.
 

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