User:Richard Pinch/sandbox-WP

Essential subgroup

A subgroup that determines much of the structure of its containing group. The concept may be generalized to essential submodules.

Definition

A subgroup \(S\) of a (typically abelian) group \(G\) is said to be essential if whenever H is a non-trivial subgroup of G, the intersection of S and H is non-trivial: here "non-trivial" means "containing an element other than the identity".

References

• Template:Cite book

Pinch point

A pinch point or cuspidal point is a type of singular point on an algebraic surface. It is one of the three types of ordinary singularity of a surface.

The equation for the surface near a pinch point may be put in the form

\[ f(u,v,w) = u^2 - vw^2 + [4] \, \]

where [4] denotes terms of degree 4 or more.

References

• Template:Cite book

Residual property

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 \(h(g)\neq e\).

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 \(\phi\colon G \to H\) where H is a group with property X.

Examples

Important examples include:

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

References

• Template:Cite book

Stably free module

A module which is close to being free.

Definition

A module M over a ring R is stably free if there exist free modules F and G over R such that

\[ M \oplus F = G . \, \]

Properties

• A projective module is stably free if and only if it possesses a finite free resolution.

See also

• Free object

References

• Template:Cite book