Proving the efficiency of pro-2-groups of fixed co-classes

Abstract

Among the six classes of pro-2-groups <br />of finite and fixed co-classes and trivial Schur Multiplicator <br />which studied by Abdolzadeh and Eick in 2009, there are two <br />classes <br />$$S_5=langle a,bmid [b,a^2]=1, a^2=[b,a]^2, <br />(b^2)^{[b,a]}b^2=1rangle$$ and $$S_6=langle a,t,bmid <br />a^2=b^2,[b,a]^2=1, t^a=t^{-1}[b,a], b^t=abarangle$$that have been <br />conjectured to have deficiency zero presentations. In this paper <br />we prove these conjectures. This completes the efficiency of all <br />six classes of pro-$2$-groups of fixed co-classes.