I'm studying Hazewinkel's "Formal Groups" book, and example of a non-commutative group law is given by:

on the ring $\mathbb{F}\_p[c]/(c^2)$ we define
$$F(X,Y)=X+Y+cXY^p.$$
I see how to prove everything except non-commutativity- my problem is I tried to find example where it fails for $p=3$ and I keep getting that this is commutative (I am making mistake somewhere):

1. I think that elements in $\mathbb{F}\_3[c]/(c^2)$ can be seen as polynomials of degree $1$ or $0$, ie. elements are of form $a\cdot c+b$.
2. Also, since characteristic of ring is $3$ we have $(a\cdot c+b)^3=a^3c^3+b^3=b$ so
3. if $F(X,Y)=F(Y,X)$ than since $X+Y=Y+X$ we must have $cXY^p=cYX^p$
4. Let $X=ac+b$ and $Y=Ac+B$ then $cXY^3=c(ac+b)(Ac+B)^3=$(step 2)$=c(ac+b)B=bBc$
5. and $cYX^3=c(Ac+B)(ac+b)^3=$(step 2)$=c(Ac+B)b=bBc$

which leads to conclusion it is commutative. Which steps are wrong?