Residual property (mathematics)
In the mathematical field of group theory, a group is residually X (where X is some property of groups) if it "can be recovered from groups with property X".
Formally, a group G is residually X if for every non-trivial element g there is a homomorphism h from G to a group with property X such that .
More categorically, a group is residually X if it embeds into its pro-X completion (see profinite group, pro-p group), that is, the inverse limit of the inverse system consisting of all morphisms from G to some group H with property X.
Examples
Important examples include:
- Residually finite
- Residually nilpotent
- Residually solvable
- Residually free
gollark: Oh, that's good, I don't think it has pOSsum.
gollark: RANDOM.
gollark: Also those 8102 digits of tau. Maybe I should compress them.
gollark: There's actually a bit of the code just listing words containing OS.
gollark: Basically, it always starts "PotatOS", then I have 8102 hardcoded digits of tau (from which it picks a random number of them), *then* a randomly picked word containing "OS", then a non-cryptographic hash of some of the code.
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