# Which finite simple groups are rational-relative-real?

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?

• 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. Jun 26, 2023 at 14:42
• 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.)
– YCor
Jun 26, 2023 at 18:25
• 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. Jun 26, 2023 at 19:17
• 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). Jun 26, 2023 at 19:27
• @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. Jun 28, 2023 at 11:09

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.