Finite group presentation ($⟨X\mid R⟩$)

A finite group presentation has finitely many generators and finitely many relators. It defines the quotient of the free group on $X$ by the normal closure of $R$. Free products of finitely presented groups and HNN extensions along explicitly finitely generated associated subgroups have finite presentations obtained by adjoining finitely many generators and relations.

A finitely presented group is a group admitting a finite group presentation. Finite presentability concerns the existence of such a presentation, not the decidability of its word problem for a group.

The Higman group has the finite group presentation $Q=⟨a_1,a_2,a_3,a_4\mid a_i{a}_{i+1}{a}_i^{-1}=a_{i+1}^2(i\mathrm{mod}4)⟩$. Its subgroup $⟨a_1,a_3⟩$ is a rank-two free group, and the normal closure of each $a_i$ is all of $Q$. It has no nontrivial finite quotients of a group: a least prime divisor among the four generator orders in a finite image contradicts the multiplicative-order constraints imposed by conjugation to squares. These properties give finite-presentation constructions embedding arbitrary finitely presented groups into groups with no nontrivial finite quotients.

A group presentation with a cyclic list of generators $a_i$ and relations $a_i{a}_{i+1}{a}_i^{-1}=a_{i+1}^2$ has no nontrivial finite quotient of a group. In a finite image, all generator orders are odd. Choose the least prime $p$ dividing any nontrivial generator order and let $x$ be the predecessor of a generator $y$ whose order is divisible by $p$. Conjugation by $x$ acts as squaring on $⟨y⟩$, so the multiplicative order of $2$ modulo $p$ divides the order of $x$. That multiplicative order is greater than one and divides $p-1$, giving a prime divisor smaller than $p$ in the order of $x$, a contradiction. This argument applies to any cycle length; it asserts absence of finite quotients, not infinitude of the presented group.

An isomorphism-invariant property of finitely presented groups is Markov when some finitely presented group has it and some finitely presented group cannot embed in any finitely presented group having it. The second witness is an obstruction to embedding, rather than merely a group failing the property. The Adian–Rabin theorem makes every such property algorithmically undecidable from arbitrary finite presentations.

No Markov property of finitely presented groups is decidable by an algorithm taking an arbitrary finite group presentation as input. The proof reduces the word problem for a group to property recognition using an effective construction which collapses when an input word is trivial and embeds the input group when it is nontrivial.