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Let $G$ be a group, $H$ a subgroup and $X$ a $G$-set. By taking orbits $X/H = X \times_H 1$ or fixed points $X^H = \mathrm{Hom}_H(1,X)$ we obtain a set on which $H$ acts trivially, and we've destroyed the $G$-action.

What we have left instead is an action of $N_H/H$, where $N_H$ denotes the normaliser of $H$, the subgroup $\{ g \in G : gH = Hg \}$.

What characterises this action abstractly?

Is there a universal property?

What other contexts does this construction exist in? Eg. rings $S \leq R$ and an $R$-module

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  • $\begingroup$ In the context of $G$-sets, 'conjugacy class of subgroups' is perhaps a more natural notion than 'subgroup', since conjugacy classes of subgroups naturally correspond to transitive $G$-sets. $\endgroup$
    – Colin Reid
    Commented Dec 16, 2014 at 0:03

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The action is the action of the (natural) automorphism group of the relevant functor in each case. This is easiest to see for the case of

$$X^H \cong \text{Hom}_H(1, X) \cong \text{Hom}_G(G/H, X)$$

since by the Yoneda lemma the automorphism group of this functor is the automorphism group of $G/H$ as a $G$-set, which is $N_H/H$. Similarly,

$$X_H \cong X \times_H 1 \cong X \times_G G/H$$

also admits a natural action by the automorphism group of $G/H$ as a $G$-set, although I am less sure if there is a clean abstract nonsense proof that these are all the natural automorphisms. There is a universal property hiding here, which is that $G/H$ is the free $G$-set on an $H$-fixed point.

Exactly the same words can be written down for endomorphism rings instead of automorphism groups in the context of rings and modules. In the context of linear representations of groups the endomorphism ring you end up writing down is a Hecke algebra.

In more general contexts it's natural to look not only at the automorphism group or the endomorphism monoid but even the endomorphism Lawvere theory or endomorphism operad. Whereas the former give unary operations, the latter gives operations of higher arity. I give some examples here and here. Other keywords: Tannaka duality, the Barr-Beck theorem...

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    $\begingroup$ I should mention that when I say "in more general contexts" I have in mind the following general question: you have a functor $F : C \to D$ and an object $c \in C$ to which you apply the functor to get an object $F(c)$. What extra structure does $F(c)$ have that reflects that it was obtained in this way? Said another way, what kind of "descent data" can we attach to $F(c)$? The starting observation is that at the very least we can attach an action of the natural automorphism group or endomorphism monoid of $F$ itself, but we can go farther than this in various ways. $\endgroup$
    – Qiaochu Yuan
    Commented Dec 19, 2014 at 21:17

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