A finite group $G$ is called *rational* if every element $g \in G$ is conjugate to all of its primitive powers $g^a, a \in (\mathbb{Z}/\operatorname{order}(g))^\times$.
Analogously, I'll call $G$ *real* if every $g$ is conjugate to its inverse. The ratio of these notions is something I'll call *rational-relative-to-real* (RRR): for every $g$, every primitive power $g^a$ should be conjugate to either $g$ or $g^{-1}$ (or both).


These conditions arise naturally in character theory. Given a field $\mathbb{k}$, let me write $\mathrm{R}\_{\mathbb{k}}(G)$ for the ring of $\mathbb{k}$-linear representations of $G$. So $\mathrm{R}\_{\mathbb{C}}(G) = \mathrm{RU}(G) = \mathrm{R}(G) $ and $\mathrm{R}\_{\mathbb{R}}(G) = \mathrm{RO}(G)$. Then $G$ is rational when $$\mathrm{R}\_{\mathbb{Q}}(G) \otimes \mathbb{Q} \to \mathrm{R}\_{\mathbb{C}}(G) \otimes \mathbb{Q}$$
is an isomorphism; $G$ is real when
$$\mathrm{R}\_{\mathbb{R}}(G) \otimes \mathbb{Q} \to \mathrm{R}\_{\mathbb{C}}(G) \otimes \mathbb{Q}$$
is an isomorphism; and $G$ is RRR when
$$\mathrm{R}\_{\mathbb{Q}}(G) \otimes \mathbb{Q} \to \mathrm{R}\_{\mathbb{R}}(G) \otimes \mathbb{Q}$$
is an isomorphism.


According to [Geoff Robinson's answer to a previous question](https://mathoverflow.net/a/449488/78), finite groups are almost never rational: any abelian composition factor has order at most 11; other than alternating groups, there are only five nonabelian simple groups that can appear.


The question I most care about it:



> 
> Which finite simple groups are RRR?
> 
> 
> 


For example, which sporadic groups are RRR?



Which finite simple groups are rational-relative-real?
Rational-relative-to-real groups are frequently called inverse semi-rational groups (or cut groups). Inverse semi-rational groups are a subclass of semi-rational groups. A finite group $G$ is *semi-rational* if for every $g \in G$ there exists a positive integer $m\_0$ such that every primitive power $g^a$ ($a \in (\mathbb{Z}/\operatorname{order}(g))^\times$) is conjugate to $g$ or $g^{m\_0}$. A semi-rational group is *inverse semi-rational* if we can take $m\_0 = -1$ for every $g \in G$ in the above definition. The terms semi-rational groups and inverse semi-rational groups were introduced in the article [Chillag–Dolfi, [Semi-rational solvable groups](https://doi.org/10.1515/JGT.2010.004), *Journal of Group Theory* **13**(4), 2010, 535-548]. In [Alavi–Daneshkhah, [On semi-rational finite simple groups](https://doi.org/10.1007/s00605-016-0964-3), *Monatshefte für Mathematik* **184**(2), 2017, 175-184] the authors determined the finite simple semi-rational groups.


Note that, in the statement of Theorem 1.1 of Alavi–Daneshkhah, the group $G\_2(4)$ is missing. The Tits group, the Suzuki groups, the Ree groups and the Steinberg triality groups are discarded in the proof, so should actually be removed from the list in Theorem 1.1.


The simple inverse semi-rational groups were derived in Theorem 5.1 of [Bächle-Caicedo-Jespers-Maheshwary, [Global and local properties of finite groups with only finitely many central units in their integral group ring](https://doi.org/10.1515/jgth-2020-0165), *Journal of Group Theory* **24**, 2021, 1163-1188].


