# Definition:Complex (Group Theory)

## Definition

Let $G$ be a group.

Let $K \subseteq G$ be a subset of $G$.

Then $K$ is referred to by some sources as a **complex** of elements of $G$.

## Notation

The notation:

- $K = A + B + C + \cdots$

can be seen for a **complex** whose elements are $A, B, C, \ldots$

## Also known as

It is commonplace to refer to a **complex** in this context merely as a **subset** of $G$.

Hence the conventional language of set theory is used in this context: $K = \set {A, B, C, \ldots}$ for $K = A + B + C + \cdots$

## Historical Note

The concept of a **complex** in the context of group theory as a synonym for a **subset** of a group is old-fashioned and perhaps even idiosyncratic.

Beyond documenting the idea, $\mathsf{Pr} \infty \mathsf{fWiki}$ will not be exploring the concept, as a treatment using the concept of a **subset** is well-developed and mathematically mainstream.

## Sources

- 1964: Walter Ledermann:
*Introduction to the Theory of Finite Groups*(5th ed.) ... (previous) ... (next): Chapter $\text {II}$: Complexes and Subgroups: $10.$ The Calculus of Complexes