Residual property (mathematics) explained

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

h(g) ≠ e .

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

\phi\colon G \to H from G to some group H with property X.

Examples

Important examples include:

• Residually finite
• Residually nilpotent
• Residually solvable
• Residually free

References

• Book: Marshall Hall (mathematician)

. The theory of groups . Marshall Hall Jr . Marshall Hall (mathematician) . New York . Macmillan . 1959 . 16 .