- #1
Dassinia
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Hello I have the solution of a problem and I don't understand it
1. Homework Statement
We know that every subgroup L<S10 acts on [1, 10] := {1, 2,..., 10} by the formula π • i = π(i). Consider L the subgroup of S10 generated by the permutation p = (1, 2, 3, 4)(4, 5)(8, 9, 10).
Find the orbit and stabilizer of 2
Orb(x)={l•x ,l ∈ L }
Stab(2)={ l ∈ L , l•x=x}
Here's the solution
L=< { p=(1, 2, 3, 4)(4, 5)(8, 9, 10) = (1, 2, 3, 4,5)(8, 9, 10) } > ⊂ S10 is generated by a 5-cycle and a 3-cycle so |L|=15 because(1, 2, 3, 4,5) &(8, 9, 10) don't commute.
Orb(2) = { 1, 2, 3, 4 , 5 }
Stab(2)={ Identity, p5, p10 }I don't understand how they found the orbit. By definition Orb(2)= L • 2 := {l • 2; l ∈ L}. l • 2= l(2) and thn I don't know what to do.
For the Stabilizer why they take p5=(8,10,9) and p10=(8,9,10) ? The order of the permutation is 15 so why Stab(2) isn't { Identity, p15} ?
Thanks
1. Homework Statement
We know that every subgroup L<S10 acts on [1, 10] := {1, 2,..., 10} by the formula π • i = π(i). Consider L the subgroup of S10 generated by the permutation p = (1, 2, 3, 4)(4, 5)(8, 9, 10).
Find the orbit and stabilizer of 2
Homework Equations
Orb(x)={l•x ,l ∈ L }
Stab(2)={ l ∈ L , l•x=x}
The Attempt at a Solution
Here's the solution
L=< { p=(1, 2, 3, 4)(4, 5)(8, 9, 10) = (1, 2, 3, 4,5)(8, 9, 10) } > ⊂ S10 is generated by a 5-cycle and a 3-cycle so |L|=15 because(1, 2, 3, 4,5) &(8, 9, 10) don't commute.
Orb(2) = { 1, 2, 3, 4 , 5 }
Stab(2)={ Identity, p5, p10 }I don't understand how they found the orbit. By definition Orb(2)= L • 2 := {l • 2; l ∈ L}. l • 2= l(2) and thn I don't know what to do.
For the Stabilizer why they take p5=(8,10,9) and p10=(8,9,10) ? The order of the permutation is 15 so why Stab(2) isn't { Identity, p15} ?
Thanks