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# Group theory

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

Please refer to the Glossary of group theory for the definitions of terms used throughout group theory.

Groups are used throughout mathematics and the sciences, often to capture the internal symmetry of other structures, in the form of automorphism groups.

In Galois theory, which is the historical origin of the group concept, one uses groups to describe the symmetries of the equations satisfied by the solutions to a polynomial equation. The solvable groups are so-named because of their prominent role in this theory.

Abelian groups underlie several other structures that are studied in abstract algebra, such as rings, fields, and modules.

In algebraic topology, groups are used to describe invariants of topological spaces (the name of the torsion subgroup of an infinite group shows the legacy of this field of endeavor). They are called "invariants" because they are defined in such a way that they don't change if the space is subjected to some deformation. Examples include the fundamental group, homology groups and cohomology groups.

The concept of Lie group (named for mathematician Sophus Lie) is important in the study of differential equations and manifolds; they combine analysis and group theory and are therefore the proper objects for describing symmetries of analytical structures. Analysis on these and other groups is called harmonic analysis.

In combinatorics, the notion of permutation group and the concept of group action are often used to simplify the counting of a set of objects; see in particular Burnside's lemma.

An understanding of group theory is also important in the physical sciences. In chemistry, groups are used to classify crystal structures, regular polyhedra, and the symmetries of molecules. In physics, groups are important because they describe the symmetries which the law of physics seem to obey. Physicists are very interested in group representations, especially of Lie groups, since these representations often point the way to the "possible" physical theories.

• A group has exactly one identity element.
• Every element has exactly one inverse.
• You can perform division in groups; that is, given elements a and b of the group G, there is exactly one solution x in G to the equation x * a = b and exactly one solution y in G to the equation a * y = b.
• The expression "a1 * a2 * ··· * an" is unambiguous, because the result will be the same no matter where we place parentheses.
• The inverse of a product is the product of the inverses in the opposite order: (a * b)−1 = b−1 * a−1.

These and other basic facts that hold for all individual groups form the field of elementary group theory.

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.

• 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. They are special sorts of categories.

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.

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

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

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