An abstract group $G$ may be a transitive subgroup of $S_n$ for different $n$ and even in different (non-conjugate) ways for a given $n$. Each action is described in the form $nTt$ where $n$ is the degree and $t$ is the T-number classifying the action.
In terms of the Galois correspondence, the group $G$ corresponds to a degree $n$ extension $F/K$ where $F=K(\alpha)$, and $G$ is the Galois group of the splitting field of the monic irreducible polynomial for $\alpha$. The siblings correspond to sibling fields, which are not isomorphic to $F$, yet have the same normal closure.
There can be more than one action with the same transitive classification. This corresponds to non-isomorphic fields with the same degree, Galois group, and Galois closure. In such a case, we append a letter, a, b, c, ... as a counter to distinguish them.
- Review status: reviewed
- Last edited by John Jones on 2018-07-09 22:53:40