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"][1], Studies in Logic and Fundations of Mathematics **71** (1973).


  [1]: http://www.sciencedirect.com/science/bookseries/0049237X/71