...... Let Ds = {r, s | r4 = e, s2 = e,rs - sr-1} be the dihedral group of order
ID: 2970195 • Letter: #
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Let Ds = {r, s | r4 = e, s2 = e,rs - sr-1} be the dihedral group of order 8 and let N = (r2). List the left cosets of N in Ds as N, rN, sN, and srN. Label these cosets with the integers 1, 2, 3, 4, respectively. Let piN : Ds rightarrow S4 be the representation obtained from the action of D% by left multiplication on the set of the four left cosets of N in D$. Show that this representation is not faithful and piN(Ds) is isomorphic to the Klein 4-group. Hint. Find piN(g) for g Ds. For example PiN(s)=[1 2 3 4 3 4 1 2] = (1,3)(2,4) because s. N=sN,s.rN=srN,s.sN=s2N=N, s.srN=s2rN=rN. Find the kernel of piN.Explanation / Answer
S3={e, (1 2),(1 3),(2 3),(1 2 3),(1 3 2)}
We need to compute (sigma(i),sigma(j)) for each sigma of S3.
orb(1,1) = { (1,1) , (2,2), (3,3) , (1,1),(2,2),(3,3)} = { (1,1) , (2,2), (3,3) }
orb(1,2) = { (1,2), (2,1),(3,2),(1,3),(2,3),(3,1)}
orb(1,3) = { (1,3), (2,3),(3,1),(1,2),(2,1),(3,2)} = orb(1,2)
orb(2,1) = { (2,1), (1,2), (2,3), (3,1),(3,2),(1,3)}=orb(1,2)
orb(2,2) = { (2,2),(1,1),(2,2),(3,3),(3,3),(2,2)}=orb(1,1)
orb(2,3) = { (2,3),(1,3),(2,1),(3,2),(3,1),(1,2)}=orb(1,2)
orb(3,1) = { (3,1),(3,2),(1,3),(2,1),(1 2),(2 3)}=orb(1,2)
orb(3,2) = { (3,2),(3,1),(1,2),(2,3),(1,3),(2,1)}=orb(1,2)
orb(3,3) = { (3,3),(3,3),(1,1),(2,2),(1,1),(2,2)}=orb(1,1)
You see there is only 2 different orbits.
let's take (1,1) in orb(1,1). We need to find a permutation such that s(1)=1
Stab(1,1)={ e, (2 3)}
Stab(2,2)={ e, (1 3)}
Stab(3,3)={e, (1 2)}
let's take (1,3) in orb(1,2). We need s(1)=1 and s(3)=3.
Stab(1,3)= { e }
We see any other stabilizer of elements in orb(1,2) is the same since fixing 2 elements fix implicitly the 3rd one.
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