Adian–Rabin theorem

In the mathematical subject of group theory, the Adian–Rabin theorem is a result which states that most "reasonable" properties of finitely presentable groups are algorithmically undecidable. The theorem is due to Sergei Adian (1955) and, independently, Michael O. Rabin (1958).

A Markov property P of finitely presentable groups is one for which:

For example, being a finite group is a Markov property: We can take ${\displaystyle A_{+}}$ to be the trivial group and we can take ${\displaystyle A_{-}}$ to be the infinite cyclic group ${\displaystyle \mathbb {Z} }$.

In modern sources, the Adian–Rabin theorem is usually stated as follows:

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