Group theory

Group theory is that branch of mathematics concerned with the study of groups.

Generalizations

In abstract algebra, we get some related structures which are similar to groups by relaxing some of the axioms given at the top of the article.

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• If we eliminate the requirement that every element have an inverse, then we get a monoid.
• If we additionally do not require an identity either, then we get a semigroup.
• Alternatively, if we relax the requirement that the operation be associative while still requiring the possibility of division, then we get a loop.
• If we additionally do not require an identity, then we get a quasigroup.
• If we don't require any axioms of the binary operation at all, then we get a magma.

Groupoids, which are similar to groups except that the composition a * b need not be defined for all a and b, arise in the study of more involved kinds of symmetries, often in topological and analytical structures.

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They are special sorts of categories.

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Supergroups and Hopf algebras are other generalizations.

Related Topics:
Supergroups - Hopf algebra

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Lie groups, algebraic groups and topological groups are examples of group objects: group-like structures sitting in a category other than the ordinary category of sets.

Related Topics:
Lie groups - Algebraic group - Topological group - Group object - Category

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Abelian groups form the prototype for the concept of an abelian category, which has applications to vector spaces and beyond.

Related Topics:
Abelian category - Vector space

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Formal group laws are certain formal power series which have properties much like a group operation.

Related Topics:
Formal group law - Formal power series

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