An Abelian group is a foundational concept in algebra that illustrates how elements combine commutatively.

The group leader assigned extra tasks to ensure the project stayed on schedule.

After his hospital discharge, he moved into a group home where nurses are available around the clock.

Several group homes in the neighborhood collaborated to organize a community festival.

The cycle group of degree k consists of all k-chains whose boundary is zero.

The boundary operator maps each k-dimensional chain group element to an element of the (k-1)-dimensional chain group, revealing cycles and boundaries in the simplicial complex.

To compute the homology group of a simplicial complex, one forms the quotient of the k-th cycle group by the k-th boundary group derived from its chain complex.

In the chain complex, the boundary group consists of all k-chains that arise as boundaries of (k + 1)-chains.

The designers used a wallpaper group to design the repeating tile so it would exhibit the desired symmetry.

The rookie boy group released a charity single to support the flood relief efforts.