1-Compute-in-S-a-a-1-ba-where-a-1-2-1-3-5-b-1-5-7-1-2-Given-permutation-1-2-3-4-1-3-5-6-Find-a-permutation-x-S-6-x-help-

Question Number 192342 by Mastermind last updated on 15/May/23

1) Compute in S_{a}, a^{- 1} ba where

a = (1 2) (1 3 5), b = (1 5 7 1)

2) Given permutation α = (1 2) (3 4),

β = (1 3) (5 6) . Find a permutation

x \in S_{6} \exists α x = β .

help!

Answered by aleks041103 last updated on 15/May/23

1)

b = (1 5 7 1) = (1 5 7)

a = (1 2) (1 3 5) = (2 1) (1 3 5) = (2 1 3 5)

\Rightarrow a^{- 1} = (5 3 1 2)

\Rightarrow a^{- 1} b a =

= (5 3 1 2) (1 5 7) (2 1 3 5) =

= (\begin{matrix} 1 2 3 4 5 6 7 \\ 1 3 7 4 5 6 2 \end{matrix}) = (2 3 7) = a^{- 1} b a

2)

α x = β, α, β, x \in S_{6}, x = ?

S i n c e S_{6} i s a g r o u p, t h e n \exists α^{- 1}

\Rightarrow α^{- 1} α x = x = α^{- 1} β

α = (1 2) (3 4)

\Rightarrow α^{- 1} = (4 3) (2 1) = (3 4) (1 2) = (1 2) (3 4) = α

\Rightarrow x = α β = (1 2) (3 4) (1 3) (5 6) =

= (1 2) (4 3) (3 1) (5 6) =

= (1 2) (4 3 1) (5 6) =

= (2 1) (1 4 3) (5 6) =

= (2 1 4 3) (5 6) =

= (1 4 3 2) (5 6) = x

Commented by Mastermind last updated on 18/May/23

Thank you boss, you really tried