I don't know how to prove that a left $KG$-module is a right $KG$-module.
What I have so far is that a left $G$-module is a right $G$-module always defining the operation $x\cdot'g=g^{-1}\cdot x$. But I don't know how to turn this to linear combinations of $KG$ since every linear combination has a element of $K$, not only elements from $G$. Any idea is highly appreciated! Thanks in advance!!