Need examples of Euclidean isometries f and g such that $\displaystyle f\circ{g}$ ≠ $\displaystyle g\circ{f}$.
let A be the matrix [cos90 -sin90|sin90 cos90] a rotation of 90. let X=(x,y) vector.
let f(X)=AX so the action of f on X rotates it by 90.
and let g be a rotation with respect to the point (1,0) not the origin.
so what g does is
g(X)=A(x-1,y)+(1,0) (i.e. take 1 away from x then rotate w.r.t origin then replace x)
so (fog)(X)=f(g(x))=A(A(x-1,y)+(1,0))
and (gof)(X)=g(f(x))=A(AX-(1,0))+(1,0)
picking point (2,1)
fog(X)=(-1,0)
gof(X)=(-1,-2)
so check by putting values for the vector X and seeing these two arent always equal.
((x-1,y) and similar expressions are vectors)