# group presentation and the inverse elements of the generators

If $G = \langle g_1 , g_2 , \ldots \mid r_i (g_1 , g_2 , \ldots ) = 1 \rangle$ is a group presentation, then is it the case that $\langle g_1^{-1} , g_2^{-1} , \ldots \mid r_i (g_1^{-1} , g_2^{-1} , \ldots ) = 1 \rangle$ is also a group presentation for $G$?

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Formally, in a group presentation $\langle x_1,\ldots,x_n \mid r_1, \ldots ,r_m \rangle$, the symbols $x_i$ on the left are not group elements, they are just abstract symbols that map onto group elements in the group defined by the presentation.

So, when you change $\langle g_1,g_2,\ldots \mid r_i(g_j) \rangle$ to $\langle g_1^{-1},g_2^{-1},\ldots \mid r_i(g_j^{-1}) \rangle$, you are not changing the group at all, you are just changing the symbols used. Just like $\langle a,b \mid a^2,b^3 \rangle$ defines the same (or isomorphic) group as $\langle c,d \mid c^2,d^3 \rangle$.

On the other hand $g_1^{-1}$ seems a very strange choice for a symbol! You would have to write its inverse as $(g_1^{-1})^{-1}$. I guess it would be even more confusing if you decided to call a generator $x^2$.

If $G\cong \langle X; \mathbf{r}\rangle$ is a presentation for $G$ and $\phi\in \operatorname{Aut}(X)$ is a Nielsen transformation then $\langle X; \mathbf{r}\phi\rangle$ is also a presentation for $G$ (here, $\mathbf{r}\phi=\{R\phi: R\in\mathbf{r}\}$). As $x\mapsto x^{-1}$ for all $x\in X$ is a Nielsen transformation, this answers your question.

Interestingly, if $G\cong \langle a, b; R(a, b)^n\rangle$ and $n>1$, so $G$ is a two-generator, one-realtor group and the relator is a proper power, then (assuming $G\not\cong \mathbb{Z}\ast C_n$) $\langle x, y; S(x, y)^m\rangle$ is a presentation for $G$ if and only if $n=m$ and there exists a Nielsen transformation of $x, y$, $\phi$ say, such that $R(x, y)=S(x, y)$. That is, you do not need any more Tietze transformations other than the basic Nielsen ones!

Another related topic is the Andrews-Curtin conjecture. This conjectures that if $\langle X; \mathbf{r}\rangle=\langle x_1, x_2, \ldots, x_n; R_1, R_2, \ldots, R_n\rangle$ is a balanced presentation (that is, $|X|=|\mathbf{r}|<\infty$) then there exists a Nielsen transformation of $X$, $\phi$ say, such that $R_i\phi\equiv x_i$. I believe it is commonly thought to be false, but is a rather important conjecture, with connections to knot theory and other fancy stuff.

If the $g_i$’s have the same interpretation in both presentations, you can deduce the presentation $\langle g_1^{-1},g_2^{-1},… \mid r_i(g_1^{-1},g_2^{-1},…)=1 \rangle$ from $\langle g_1, g_2,… \mid r_i(g_1,g_2,…)=1 \rangle$ using Tietze transformations.