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∈S


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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

	$Thank you boss, you really tried$
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