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feat(Algebra/Group/Subgroup, GroupTheory/QuotientGroup): strengthen second isomorphism theorem for groups #22650
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PR summary a90a44f49cImport changes for modified filesNo significant changes to the import graph Import changes for all files
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Co-authored-by: Paul Lezeau <[email protected]>
The stronger statement is necessary when we wish that
H
andH ⊔ N
in the statement_ ⧸ N.subgroupOf H ≃* _ ⧸ N.subgroupOf (H ⊔ N)
may need to be of a type larger than whichever subgroupH
is normal in. Furthermore, this commit adapts some basic results about Normal subgroups to be wrt subgroups of normalizers instead.Moves:
The primary reason for introducing this variation is that the existing formulation becomes a pain when quotienting subnormal groups. So suppose my ambient group is
G
and my normalizer isT : Subgroup G
, the result I can get from the existing formulation is something like_ ⧸ (N : T).subgroupOf (H : T) ≃* _ ⧸ (N : T).subgroupOf (H ⊔ N : T)
, and it is difficult or tedious to recover the desired statement_ ⧸ N.subgroupOf H ≃* _ ⧸ N.subgroupOf (H ⊔ N)
from it.