gh-36865: `sage.groups`, `sage.rings.number_field`: Modularization fixes, `# needs`
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- Cherry-picked from #35095
Standard modularization work:
- a new dynamically detected feature `sage.libs.braiding`
- using file-level and block-level `# needs` to simplify decorations
- adding some `# needs`, in particular for `sage.libs.gap`
- splitting GAP-dependent classes out from `sage.groups.galois_group`
- using more `lazy_import`s
- some doctest cosmetics
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Part of:
- #29705
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### :hourglass: Dependencies
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URL: https://github.com/sagemath/sage/pull/36865
Reported by: Matthias Köppe
Reviewer(s): David Coudert
