TITLE: Is it decidable to check if an element has finite order or not?
QUESTION [36 upvotes]: Suppose we have a finitely presented group $G$ with decidable word problem. Is it decidable to check whether a given element $x\in G$ has finite order or infinite?

REPLY [21 votes]: The decidability of the word problem does not imply the decidability of the order problem, and in fact the following more general result holds.

Theorem. Let $\mathbf{a}, \, \mathbf{b}, \, \mathbf{c}$ be three recursively enumerable degrees of unsolvability (i.e., Turing degrees) with $\mathbf{a} \leq \mathbf{b}$ and $\mathbf{a} \leq \mathbf{c}$. Then there exists a finitely presented group $L$ such that 

the word problem for $L$ is of degree $\mathbf{a};$
the power problem for $L$ is of degree $\mathbf{b};$
the order problem for $L$ is of degree $\mathbf{c}.$


See 
D. J. Collins: The word, power and order problems in finitely presented groups, in "Word Problems, Decision Problems and the Burnside Problem in Group Theory", Studies in Logic and Fundations of Mathematics 71 (1973).