PROVING THE EFFICIENCY OF PRO-2-GROUPS OF FIXED CO-CLASSES

Among the six classes of pro-2-groups of finite and fixed co-classes and trivial Schur Multiplicator which studied by Abdolzadeh and Eick in 2009, there are two classes S-5=langle a,bmid [b,a^2]=1, a^2=[b,a]^2, (b^2)^{[b,a]}b^2=1rangle and S-6=langle a,t,bmid a^2=b^2,[b,a]^2=1, t^a=t^{-1}[b,a], b^t=aba rangle that have been conjectured to have deficiency zero presentations. In this paper we prove these conjectures. This completes the efficiency of all six classes of pro-$2$-groups of fixed co-classes.