Glossary of Group Theory

Glossary Of Group Theory

A group (G, •) is a set G closed under a binary operation • satisfying the following 3 axioms:

• Associativity: For all a, b and c in G, (a • b) • c = a • (b • c).
• Identity element: There exists an e∈G such that for all a in G, e • a = a • e = a.
• Inverse element: For each a in G, there is an element b in G such that a • b = b • a = e, where e is an identity element.

Basic examples for groups are the integers Z with addition operation, or rational numbers without zero Q\{0} with multiplication. More generally, for any ring R, the units in R form a multiplicative group. See the group article for an illustration of this definition and for further examples. Groups include, however, much more general structures than the above. Group theory is concerned with proving abstract statements about groups, regardless of the actual nature of element and the operation of the groups in question.

This glossary provides short explanations of some basic notions used throughout group theory. Please refer to group theory for a general description of the topic. See also list of group theory topics.