Please do the solution within 1 hour,

1. (a) Let F be a free group over {x, y} C F. i) Show that U = (xay : a, b E Z } is a Schreier transversal for the commutator subgroup [F, F] in F. [4 marks] ii) Write down the Schreier generators for [ F, F] corresponding to the Schreier transversal U from i). [4 marks] iii) Express the element xyx y xE [F, F] in terms of the Schreier generators from ii ) . [4 marks] (b) Let G be the group defined by the presentation G = (x, y, zylxy= y*-2x-lyk+2, z-lyz = zm-2x-Izm+2 , x zx = x-2z-1x7+2), where k, m, n are non-zero even integers. i) Use the first relation to show that xy = y x. [3 marks] ii) Deduce that H = (x , y , z ) is an abelian subgroup of G. [5 marks] iii) Show that ux u E H for each u E (x, x , y, y , z, z ). What can you deduce from this about H? [5 marks]