# How can a field L be a subgroup of the Galois group of the extension L:K?

I am studying the notion of a Galois group, and a remark in my notes is the following:

"$$L:K$$ an extension. Let $$H$$ be a subgroup of $$Aut(L)$$. Let $$L^{H}$$ be the fixed field of $$H$$.

If $$L \leq G(L:K)$$, then $$L^{H}$$ is an intermediate field of $$L:K$$. "

Now, I don't understand how we can have $$L \leq G(L:K)$$. $$L$$ is a an arbitrary field, but the group $$G(L:K)$$ is a group of $$K$$-automorphisms of $$L$$. How can the entire field $$L$$ be subgroup of a group of automorphisms of $$L$$?

This seems totally strange to me, as automorphisms are in general different objects to elements of $$L$$ so how can we even talk about them being subgroups of each other?

• It's probably a typo and they meant to say "If $H \leq G(L : K)$, then $L^H$ is an intermediate field of $L : K$."
– D_S
Apr 10 '20 at 18:33
• Yeah this is what I thought. Thanks! Apr 10 '20 at 19:09