Assuming I have two elements of the permutation group algebra $a_1$, $a_2$ such that $a_i=\sum_{p\in S_n}\alpha_pp$, I want to define a linear product $\odot$ that distributes over the sum and pulls out scalars: $a_1\odot a_2=\sum_{p\in S_n}\sum_{q\in S_n}\alpha_p\alpha_qp\odot q$, where $p\odot q\equiv PermutationProduct[p,q]$. If I try for example
Distribute[PermutationProduct[Cycles[{{}}] + Cycles[{{1, 2}}], Cycles[{{}}] - Cycles[{{1, 2}}]]}
The result is
Cycles[{}] +PermutationProduct[Cycles[{{1, 2}}],(-Cycles[{{1, 2}}])]
but I'm not sure how to make it take "constants" outside the product.