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

Document Type: Research Paper

Authors

Abstract

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 $$S_6=langle a,t,bmid
a^2=b^2,[b,a]^2=1, t^a=t^{-1}[b,a], b^t=abarangle$$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.

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