Does the parity of the identity of a group hold any known significance?

Question

I understand that for permutation groups, the parity of identity is even.

However when considering, for example $\langle\mathbb{Q} \setminus \{0\}, \cdot\rangle$ where $\cdot$ is regular multiplication, the identity is odd.

Does the parity of the identity tell us anything about the group or the operation?

Answer 1

The two notions of parity you've brought up are really unrelated. In fact the "parity" of $1 \in \mathbb{Z}$ is not intrinsic to that group in any way. You could have labeled the elements of $\langle \mathbb{Z}, \cdot \rangle$ however you liked, not even necessarily with numbers, as long as the binary operation works out the same way. You could have called the multiplicative identity $\mathfrak{I}$ if you wanted, or $\ddot\smile$. What would the parity of the identity be then?

Answer 2

By Cayley's theorem, each group $G$ is isomorphic to a subgroup $H$ of a symmetric group. Each subgroup of a group is itself a group. The parity of the identity in $H$ is always even (as a permutation).