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