Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 104 / 6 / d

Every finitely presented group $P$ embeds in $P\times C_2$ by $p\mapsto (p,1)$. This direct product of groups is finitely presented: combine finite presentations of the factors and add the finitely many cross-commutation relations between their generators. Projection onto $C_2$ gives a nontrivial finite quotient of a group.

Thus no finitely presented $P$ can serve as the forbidden witness $A_{-}$ for this property. Hence

Part (c) shows that every finitely presented group embeds in a finitely presented group with no nontrivial finite quotients of a group. Consequently this property also has no possible forbidden witness $A_{-}$. ThereforeBoth properties have positive witnesses, for example $C_2$ and the trivial group respectively. It is the required embedding obstruction that fails in each case.