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Groups

Let $G$ be a finite group. An involutive automorphism on $G$ is an automorphism $i\colon G \to G$ such that $i^2 = 1_G$.

Question 1. What classifies involutive automorphisms on a given (non-abelian) finite group? In particular, is there any classification in terms of group cohomology?

Note that the involutive automorphisms needn't form a group, just like the involutions (the "inner involutive automorphisms", if you wish) don't necessarily.

Observe the following: Whenever $h = i(h)^{-1}$, then $j(g) = h^{-1}i(g)h$ is again involutive.

Question 1.1. Is the set of involutive automorphisms a torsor for some group?

Given two involutive automorphisms $i$ and $j$, then $ji$ is an automorphism of $G$, but I don't see any guarantee that it is inner, let alone conjugation by an element $h$ satisfying $i(h) = h^{-1}$. Now famously the number of involutions is given by a formula involving irreducible characters and Frobenius-Schur indicators, but I'm not aware of any such result for involutive automorphisms.

Fusion rings

Let $R$ be a unital ring with a finite basis $B$ containing the neutral element such that all structure constants are nonnegative. Such rings are called based rings. Morphisms of based rings are required to map basis elements onto basis elements.

A fusion ring is a ring structure on $\mathbb{Z}^n$ with basis $\{(1,0,\dots), (0,1,0,\dots), \dots\}$, together with an anti-involution $i$ on $R$ such that for every basis element $b$, the multiplicity of the neutral element in $b \cdot i(b)$ is exactly $1$. (The Grothendieck ring of a fusion category is always a fusion ring, hence the name. The extra condition on the chosen anti-involution encodes the existence of duals in the category.)

It is easy to check that every finite group gives rise to a fusion ring, where the chosen anti-involution is inversion. But there are also many fusion rings that don't come from groups, for example the Fibonacci ring with basis $\{1, \tau\}$ satisfying $\tau \cdot \tau = 1 + \tau$.

Questions 2-2.1. What classifies involutive automorphisms on fusion rings? (And the same further questions as for groups.)

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  • $\begingroup$ For question 1.1, what do you mean by "torsor"? The identity is an involutive automorphism, so if a (right) torsor structure were compatible with composition of automorphisms (on the left), the involutive automorphisms would have to be a group under composition. $\endgroup$ – MTyson Sep 13 '17 at 3:47
  • $\begingroup$ @MTyson, it couldn't be compatible with the composition of automorphisms, since two involutions don't compose to an involution. But thanks for the comment, I added two links to hopefully clarify the situation. $\endgroup$ – Manuel Bärenz Sep 13 '17 at 8:44
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    $\begingroup$ I don't really understand what would constitute an answer to this question. Let's consider replacing involutions with just automorphisms. What would you consider an answer to the question "what classifies automorphisms of groups?" Note that there is no hope of something like a "classifying space" or anything resembling group cohomology because taking automorphisms is not functorial, and the same comment applies to involutions. $\endgroup$ – Qiaochu Yuan Sep 13 '17 at 8:55
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    $\begingroup$ @QiaochuYuan, that's already a valuable insight that I didn't have. Since the number of (inner) involutions of a group can be calculated, and a set by itself doesn't have any more structure than its number of elements, one could hope for a formula that tells us the number of involutive automorphisms. Or there could be a partial answer like "assume we know the group of automorphisms, then the involutions are classified by the following thing". $\endgroup$ – Manuel Bärenz Sep 13 '17 at 10:00

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