# Normalizers of automorphism groups

In abstract groups $\Gamma$ the normalizer $N_\Gamma(S)$ of a subset $S\subseteq\Gamma$ is the subgroup of all $x \in \Gamma$ that commute with $S$, i.e. $xS = Sx$, i.e. $x\ y\ x^{-1} \in S$ for all $y \in S$.

Among the permutations $S_n$ of the vertices of a graph $G$ of order $n$ (or any other kind of structure) there is one distinguished subgroup: the automorphisms $\text{Aut}(G)$, that reflect the symmetries of $G$:

$$\alpha \in \text{Aut}(G)\quad \Leftrightarrow \quad \alpha G = G$$

To give $\alpha G = G$ a proper meaning, identify $G$ with an adjacency matrix, for example.

The normalizer of $\text{Aut}(G)$ is another distinguished subgroup: it consists of those permutations $\pi$ of the vertices, such that $\text{Aut}(\pi G) = \text{Aut}(G)$, i.e. that respect the symmetries of $G$, as can be shown like this:

$\quad \pi \in N_{S_n}(\text{Aut}(G))\\ \Leftrightarrow \pi^{-1}\alpha\ \pi \in \text{Aut}(G)\ \text{for all}\ \alpha \in \text{Aut}(G)\\ \Leftrightarrow \pi^{-1}\alpha\ \pi\ G = G\ \text{for all}\ \alpha \in \text{Aut}(G)\\ \Leftrightarrow \alpha\ \pi\ G = \pi\ G\ \text{for all}\ \alpha \in \text{Aut}(G)\\ \Leftrightarrow\alpha \in \text{Aut}(\pi G)\ \text{for all}\ \alpha \in \text{Aut}(G)$

Note that $\text{Aut}(\pi G)$ and $\text{Aut}(G)$ are of course isomorphic for every $\pi \in S_n$:

$$\text{Aut}(\pi G) \simeq \text{Aut}(G)$$

but this is not the matter of concern. The matter of concern is

$$\text{Aut}(\pi G) = \text{Aut}(G)$$

My first question now is:

Does the normalizer of the automorphisms of a structure has an established name on its own?

Something like symmetry preserving rearrangements (compared to adjaceny preserving rearrangements [what automorphisms are] or structure preserving rearrangements [what general permuations - of labels - are])? Note, that and how the following permutation is (i) symmetry preserving and (ii) an element of the normalizer of $\text{Aut}(G)$ and that (iii) most other permuations are not:

Where is the normalizer of the automorphisms of a structure investigated in its own right - or plays an explicit role, e.g. in a theorem?

More specific:

How can the normalizer of the automorphisms of a structure be defined/characterized without reference to (and prior definition of) the latter?

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This is an interesting question to which I don't have an answer. A non-mathematical remark: when there is a need to rewrite a question (esp. one with no answers), it's much better to edit than to create another copy. Otherwise the old question remains as a pollutant in the pristine sea of knowledge called MSE. – user31373 Jul 18 '12 at 4:15
I am going to delete the other question. I thought about editing it, but I would have had to change the title, and this didn't seem appropriate to me. – Hans Stricker Jul 18 '12 at 10:56
A really minor point, but unless $S\subset \Gamma$ is finite, the two conditions $$x\in N_{\Gamma}(S), \mathrm{~i.e.~} xSx^{-1}=S$$ and $$\forall s\in S, ~xsx^{-1}\in S\mathrm{~i.e.~}xSx^{-1}\subset S$$ aren't equivalent. You ask for a structure where the normalizer of a symmetry group is investigated in its own right, it is (in my modest understanding) important in the theory of covering spaces and thus also in the theory of field extensions. Under the right topological conditions, the covers of a space are classified by the subgroups $G$ of the fundamental group $\pi_1 B$ of the base space, – Olivier Bégassat Jul 18 '12 at 23:21
and given the corresponding covering $X\rightarrow B$, the symmetries of this cover correspond to the group $$N_{\pi_1B}(G).$$ I think the same holds for Galois extensions. In both cases this is part of the Galois correspondence. – Olivier Bégassat Jul 18 '12 at 23:23
I mean the quotient $N(G)/G$. – Olivier Bégassat Jul 26 '12 at 0:30