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?



Is it decidable to check if an element has finite order or not?
A finitely presented group with decidable word problem and undecidable order problem is in [McCool, James
Unsolvable problems in groups with solvable word problem. 
Canad. J. Math. 22 1970 836–838](http://cms.math.ca/10.4153/CJM-1970-094-6). 


