Dehn twists on nonorientable surfaces

Let t(a) be the Dehn twist about a circle a on all orientable surface. It is well known that for each circle b and an integer n, I(t(a)(n) (b), b) = \n\I(a, b)(2), where I(., .) is the geometric intersection number. We prove a similar formula for circles on nonorientable surfaces. As a corollary we prove some algebraic properties of twists on nonorientable surfaces. We also prove that if M(N) is the mapping class group of a nonorientable surface N, then up to a finite number of exceptions, the centraliser of the subgroup of M (N) generated by the twists is equal to the centre of M (N) and is generated by twists about circles isotopic to boundary components of N.