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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, 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?

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    $\begingroup$ There are many finite simple groups with this property. For example, looking at the Atlas, in order of size, there are $L_2(7)$, $A_7$, $U_3(3)$, $M_{11}$, $A_8$, $M_{12}$, $U_3(5)$, $A_9$, $M_{22}$, and so on. You might try looking at the Atlas for yourself for the sporadic ones. $\endgroup$ Jun 26, 2023 at 14:42
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    $\begingroup$ No, for $p$ prime, $\mathrm{Alt}(p)$ is not RRR if $p\equiv 1$ mod $4$, as $c^k$ is not conjugate to $c^{\pm 1}$ when $k$ is a multiplicative generator mod $p$ and $c$ is a $p$-cycle. (This is "visible" in $\mathrm{Alt}(5)$ viewed as group of motions of the icosahedron, since rotations of angle $2\pi/5$ and $4\pi/5$ are neither conjugate, nor inverse-conjugate.) $\endgroup$
    – YCor
    Jun 26, 2023 at 18:25
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    $\begingroup$ The values of an irreducible character of ${\rm Alt}(n)$ at an element of cycle-type $\lambda$ are all rational, unless the parts of $\lambda$ are all odd, and none are repeated, in which case, the character values lie in $\mathbb{Q}[ \sqrt{\epsilon \prod_{i} \lambda_{i}]},$ where $\lambda_{i}$ is the $i$-th part of $\lambda$, and the product is congruent to $\epsilon$ (mod $4$), This result appears in James and Kerber is used in a paper that Thompson and I wrote in 1994 in the Journal of Algebra, determining the field generated by all character values of a given alternating group. $\endgroup$ Jun 26, 2023 at 19:17
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    $\begingroup$ I think a similar phenomenon occurs in ${\rm PSL}(2,p)$ when $p \equiv 1$ (mod $4$), for then $-1$ is a quadratic residue (mod $p$), and ${\rm PSL}(2,p)$ is not $RRR$ ( this also explains ${\rm Alt}(5)$ another way). $\endgroup$ Jun 26, 2023 at 19:27
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    $\begingroup$ @LSpice : I think I was more restrained than that in my language in my answer to the previous question, but it is fair to say that symmetric groups have pretty small measure as a subset of the set of all finite groups. $\endgroup$ Jun 28, 2023 at 11:09

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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, Journal of Group Theory 13(4), 2010, 535-548]. In [Alavi–Daneshkhah, On semi-rational finite simple groups, 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, Journal of Group Theory 24, 2021, 1163-1188].

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    $\begingroup$ @LSpice thanks for adding links to the articles. $\endgroup$
    – bl'
    Jun 27, 2023 at 21:12

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