= Canonical presentation of a group
{title2=$\langle G\mid\ker(F(G)\to G)\rangle$}
= Canonical presentations of a group
{synonym}
The <canonical presentation of a group> uses one symbol for each <group element> and all words that evaluate to the <identity element> as <relators>. Evaluation is a <surjective> <group homomorphism> from the <free group> on the underlying <set>; the <first isomorphism theorem> proves that the resulting <group presentation> recovers the original <group>. Equivalently the multiplication-table <relators> $s_gs_hs_{gh}^{-1}$ normally generate its kernel: they reduce every word to the symbol for its value. https://people.maths.ox.ac.uk/lackenby/tg11o17.pdf#page=44[Topology and Groups, Definition V.8 and preceding construction, printed p. 44]{ref}
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