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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 identified by a label of 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. We indicate multiplicity using the notation "
nTt x k" where there are
k non-conjugate subgroups such that the action is
- Review status: reviewed
- Last edited by Andrew Sutherland on 2020-10-22 07:58:06
- 2020-10-22 07:58:06 by Andrew Sutherland (Reviewed)
- 2020-10-22 07:55:47 by Andrew Sutherland
- 2019-05-29 14:42:34 by John Jones
- 2018-07-09 22:53:40 by John Jones (Reviewed)