groupshave

Groupshave is a term used in group theory and related fields to describe a notion that there may be shared invariants across a collection of groups acting on a common set. It is not a single, standardized concept, but rather a family of loosely related ideas that emphasize common structure or fixed elements arising from multiple group actions.

One informal formulation states that a collection of groups {G_i} acting on a set X has the groupshave property if there exists a nonempty finite subset F of X that is fixed pointwise by every G_i. In other words, every element of F remains unchanged under the action of every group in the family. This viewpoint highlights shared invariants or simultaneous symmetries rather than properties of any individual group.

Variations of the idea relax or alter the conditions, such as allowing F to be infinite, replacing fixed points with common orbits, or focusing on common kernels of the actions. Some presentations use the term to discuss conditions under which a family of actions admits a nontrivial common fixed set or a common invariant substructure.

Origin and usage of groupshave are informal and typically appear in expository writings, lecture notes, or discussions intended to illustrate how multiple symmetry structures interact. Because the term lacks a formal, widely adopted definition, precise usage depends on context and author.

See also: group action, fixed point, invariants, symmetry group, orbit-stabilizer.