TITLE: Are there any interesting theorems of first-order group theory
QUESTION [7 upvotes]: I've heard that theoremhood in first-order group theory is uncomputable, so there should be some theorems that are difficult to prove. However the few theorems of group theory I know are about subgroups, exponents and such, so they don't seem to be formalizable in first-order logic (for each $n\in\mathbb{N}$ we can write $\forall_x x^n=e$ for "the group has exponent $n$" but this doesn't seem to help much).
Are there any cool theorems or conjectures that are naturally stated in the first-order language of group theory?

REPLY [5 votes]: It is proved in [1] that a finite group is soluble if and only if it satisfies the following first order sentence, which states that no non-trivial element $g$ is a product of $56$ commutators of pairs of conjugates of $g$:
$$
\forall g\ \forall x_1 \dotsm \forall x_{56}\ \forall y_1 \dotsm \forall y_{56}\ \bigl(g = 1 \vee g \not= [g^{x_1}, g^{y_1}] \dotsm [g^{x_{56}}, g^{y_{56}}]\bigr)
$$
[1] J. S. Wilson – Finite axiomatization of finite soluble groups, J. London Math. Soc. (2) 74 (2006), 566–582.