- #1
mathbalarka
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Consider a group $G$ and a subgroup $H$. Define $Aut_H(G)$ to be the group of automorphisms of $G$ which fixes $H$ pointwise.
Claim : If $N \leq H \leq G$ and $N$ is a characteristic subgroup of $H$, which in turn is a characteristic subgroup of $N$, then there is a left exact sequence of groups
$$1 \to Aut_H(G) \to Aut_N(G) \to Aut_N(H)$$
The proof is quite straightforward : Exactness of $1 \to Aut_H(G) \to Aut_N(G)$ is more or less obvious, only nontrivial part being exactness at the fourth morphism. This is accomplished as follows : Take the map $\varphi : Aut(G) \to Aut(H)$ induced by restriction of the automorphisms to $H$. Note that this is well defined as $H$ is a characteristic subgroup of $G$. Now restriction of the domain of $\varphi$ to $Aut_N(G)$ induces the map $\varphi' : Aut_N(G) \to Aut_N(H)$ as $N$ is a subgroup of $H$. The kernel of this map are the automorphisms inside $Aut_N(G)$ fixing $H$ pointiwse, which is precisely $Aut_H(G)$. Thus we are done.
I am sure the reader has not failed to note that $Aut_H(G)$ looks much like the Galois group at this point, and the claim above is a plain mimicry of the short exact sequence of Galois groups $1 \to Gal(K/E) \to Gal(K/F) \to Gal(E/F) \to 1$. Furthermore, Galois extensions are also analogized by characteristic subgroups. In the whole, from what is apparent, the Galois theory has been redefined here in the context of groups - an object of much less structure than a field.
As far as I have known, all the remarkable analogies of Galois theory lies in covering spaces, so I don't believe this is of much interest. What's fun is that the corresponding $\mathbf{Fix}$ functor here translates short exact sequences $1 \to A \to B \to C \to 1$ to left-exact sequences $1 \to \mathbf{Fix}(A) \to \mathbf{Fix}(B) \to \mathbf{Fix}(C)$. This is one of the key observations of group cohomology theories. Also, note that the sequence of automorphisms groups in the claim is not right-exact, which gives a faint hope of extending it to the right by some clever use of category theory. More to the point, I believe much fiddling can be done with it but most of them would require knowledge beyond my grasp. So any ideas, suggestions, comments or clarifications are most welcome.
Balarka
.
Claim : If $N \leq H \leq G$ and $N$ is a characteristic subgroup of $H$, which in turn is a characteristic subgroup of $N$, then there is a left exact sequence of groups
$$1 \to Aut_H(G) \to Aut_N(G) \to Aut_N(H)$$
The proof is quite straightforward : Exactness of $1 \to Aut_H(G) \to Aut_N(G)$ is more or less obvious, only nontrivial part being exactness at the fourth morphism. This is accomplished as follows : Take the map $\varphi : Aut(G) \to Aut(H)$ induced by restriction of the automorphisms to $H$. Note that this is well defined as $H$ is a characteristic subgroup of $G$. Now restriction of the domain of $\varphi$ to $Aut_N(G)$ induces the map $\varphi' : Aut_N(G) \to Aut_N(H)$ as $N$ is a subgroup of $H$. The kernel of this map are the automorphisms inside $Aut_N(G)$ fixing $H$ pointiwse, which is precisely $Aut_H(G)$. Thus we are done.
I am sure the reader has not failed to note that $Aut_H(G)$ looks much like the Galois group at this point, and the claim above is a plain mimicry of the short exact sequence of Galois groups $1 \to Gal(K/E) \to Gal(K/F) \to Gal(E/F) \to 1$. Furthermore, Galois extensions are also analogized by characteristic subgroups. In the whole, from what is apparent, the Galois theory has been redefined here in the context of groups - an object of much less structure than a field.
As far as I have known, all the remarkable analogies of Galois theory lies in covering spaces, so I don't believe this is of much interest. What's fun is that the corresponding $\mathbf{Fix}$ functor here translates short exact sequences $1 \to A \to B \to C \to 1$ to left-exact sequences $1 \to \mathbf{Fix}(A) \to \mathbf{Fix}(B) \to \mathbf{Fix}(C)$. This is one of the key observations of group cohomology theories. Also, note that the sequence of automorphisms groups in the claim is not right-exact, which gives a faint hope of extending it to the right by some clever use of category theory. More to the point, I believe much fiddling can be done with it but most of them would require knowledge beyond my grasp. So any ideas, suggestions, comments or clarifications are most welcome.
Balarka
.