Question

Let$\mathbf{\alpha }$ and$\mathbf{\beta }$ be permutations in S_n

1. Fill the blanks in the table.

$\mathbf{\beta }$	$\mathbf{\alpha }$	${\mathbf{\alpha \beta \alpha }}^{\mathbf{-}\mathbf{1}}$	${\mathbf{\alpha \beta \alpha }}^{\mathbf{-}\mathbf{1}}{\mathbf{\beta }}^{\mathbf{-}\mathbf{1}}$
even	even
odd	even		even
even	odd
odd	odd

Short answer

$\mathrm{\beta }$	$\mathrm{\alpha }$	${\mathrm{\alpha \beta \alpha }}^{-1}$	${\mathrm{\alpha \beta \alpha }}^{-1}{\mathrm{\beta }}^{-1}$
even	even	even	even
even	odd	odd	even
odd	even	even	even
odd	odd	odd	even

Step-by-step solution

Transposition

If$\mathrm{\alpha }$ and$\mathrm{\beta }$ are two element, and$\mathrm{\alpha }$ can be written as a product of p transposition and$\mathrm{\beta }$ as a product of q transposition, then their product$\mathrm{\alpha \beta }$ can be written as a product of p+q transposition.

Answer

Therefore, the complete table is:

$\mathrm{\alpha }$	$\mathrm{\beta }$	${\mathrm{\alpha \beta \alpha }}^{-1}$	${\mathrm{\alpha \beta \alpha }}^{-1}{\mathrm{\beta }}^{-1}$
even	even	even	even
even	odd	odd	even
odd	even	even	even
odd	odd	odd	even