Give an example of elements a,b,c,d and x from a group where axb=cxd but( ab) not equal( cd)?

middle cancellation is not valid in groups.

Give an example of elements a,b,c,d and x from a group where axb=cxd but( ab) not equal( cd)?
Consider the dihedral group of order 6, with generators f and r, satisfying f² = e, r³ = e, and fr = r⁻¹f. Let a=r, b=r, c=e, d=e, and x=f. Then we have:





axb = rfr = rr⁻¹f = f


cxd = efe = f





However:





ab = rr = r² ≠ e = ee = cd.
Reply:Hey... you can use vector analysis!!!





you see in vector analysis...





(ab),(cd) are called dot products which is scalar quantity





while (axb), (cxd) are cross products which is vectoral quantity.





try to read books about vector analysis or engineering electromagnetics.