Group action Invariants (reviewed)

If a group $G$ acts transitively on an $n$ element set, the action is isomorphic to the natural action of a transitive subgroup of $S_n$. In terms of the Galois correspondence, we consider an irreducible separable polynomial $f(x)\in K[x]$, let $F$ be a field obtained by adjoining a root of $f(x)$ to $K$ and suppose $G$ is the Galois group of the splitting field of $f(x)$ over $K$ where the action on the roots of $f(x)$, $\alpha_1, \ldots, \alpha_n$ is through the subscripts.

The degree is $n$, the number of elements in the set which equals $\deg(f)=[F:K]$.

Transitive subgroups of $S_n$ have been classified and numbered; $t$ is the corresponding number assigned to this group.

The parity is $1$ if $G$ is a subgroup of $A_n$, otherwise it is $-1$.

The action is primitive if the only block of size larger than 1 for the action is the whole set. In terms of the Galois correspondence, the action of $G$ is primitive if and only if $F/K$ has no proper intermediate fields.

Resolvent fields are other fields inside the splitting field of $f(x)$ over $K$. These correspond to small index subgroups of $G$,up to conjugation. For a degree $m$ field $E$, the action of $G$ on the embeddings of $E$ into the splitting field of $f(x)$ gives a transitive group action, which is faithful for the appropriate quotient of $G$. These are given in terms of the degree of the normal closure of $E$, and information describing the faithful transitive group action corresponding to $E$.