Q: Let $\alpha_1 , …, \alpha_r$ be distinct even permutations, and $\beta$ an odd permutation. Show then $\alpha_1\beta, …, \alpha_r \beta$ are $r$ distinct odd permutations.
Proof: $\alpha_i$ can be written as a product of $2k$ transpositions, $\beta$ can be written as a product of $m$ odd transpositions, $\implies\alpha_i\beta$ can be written as a product of $2k+m$ transpositions, which is odd, this implies that $\alpha_i\beta$ is odd.
Now $\alpha_i\beta$ is distinct because of the fact that $\alpha_i\beta$ is equivalent to $\alpha_i$ over a different domain, that is the domain $\{(\beta(1),\cdots,\beta(n)\}$, and since all $\alpha_i$ are distinct, this implies that $\alpha_i\beta$ is distinct aswell
Is this proof good? I ask not because I struggle to write this proof, I'm actually fairly confident it is correct, I am just asking because I am a self-studying and have no mentorship in proof writing, so I've sort of just tried to pick it up along the way. Can I please have some tips as to how I could have done better on this proof? Or if it's okay already, please tell me as I am operating in the dark when it comes to these new textbooks I am using that do not have answers at all!
REVISED PROOF: $\alpha_i$ is written as a product of $2k$ transpositions, $\beta$ a product of $2m+1$ transpositions, $\implies$ $\alpha_i\beta$ is written as a product of $2k+2m+1=2(k+m)+1$ transpositions, which is odd $\implies \alpha_i\beta$ is odd. $\alpha_i\beta$ is distinct by the fact that $\alpha_i$ is distinct $\forall i$ and $\beta$ is constant.