# Decision problems and group representations

If one takes a group presentation then one can ask various questions of it, such as "is this element equal to the identity", "are these elements conjugate" etc. I was wondering if the solution to such a problem in a representation of a group always yields a solution to the problem with respect to the presentation.

For example, if $G$ is a finitely generated group with soluble word problem then one can use the word problem for $G$ to work out if two elements of $\operatorname{Aut}(G)$ are different in finite time (if $\phi: x_i\mapsto X_i$ and $\varphi: x_i\mapsto Y_i$ then $\phi=\varphi$ if and only if $X_i=Y_i$ for all $i\in I$, $|I|<\infty$). However, I am unsure whether this amounts to a solution to the word problem for $\operatorname{Aut}(G)$. This is because in order to solve the word problem for $\operatorname{Aut}(G)\cong\langle X; R\rangle$ this way one would need to know in what way the given presentation is $\operatorname{Aut}(G)$; one would need to first know the isomorphism between $\operatorname{Aut}(G)$ and $\langle X; R\rangle$, but...can this always be done?

I am expecting the answer to be "yes, of course, don't be stupid!" but I just can't see how this would hold (although obviously it should)!

• I am unsure as to how your first paragraph is illustrated by the second. To me you seem to be asking two different questions. The first paragraph sounds like it's asking "If I can solve a representation's word problem, then can I solve $G$'s word problem?" The second paragraph seems like it's asking "If I can solve $G$'s word problem, can I also solve $\mathrm{Aut}(G)$'s word problem?" – Bill Cook Sep 19 '11 at 20:01
• I was taking Aut(G) to be a representation for the abstract group Aut(G). I am not meaning linear representations, just representations in a more general sense (realisations?). – ADL Sep 20 '11 at 8:54