Let $\sigma(v)$ denote the signature of the permutation $v$. Is the subset $L_7 = \{v\in S_7 : \sigma(v)=-1\}$ a subgroup of $S_7$?
I am not sure I am proving it the right way.
To prove that $L_7$ is a subgroup first I have to show that for any $a$, $b$ from $L_7$ then $a \cdot b$ belongs to $L_7$.
Suppose $a$ and $b$ belong to $L_7$, that means $\sigma(a)=-1$ and $\sigma(b)=-1$. That means both of them can be written as a product of an odd number of transposition (let's say $k$ and $l$). Therefore, the composition of the permutations $a \cdot b$ will be written as a product of an even number of transposition (as if $k$ and $l$ are odd then $k+l$ will always be odd).
So $\sigma(a\cdot b)=1$. Then it doesn't belong to $L_7$. Therefore $L_7$ is not a subgroup of $S_7$.
I'm not sure if I am proving it the right way though. Am I doing something wrong?
