In mathematics, a finite group is a group which has finitely many elements. Some aspects of the theory of finite groups were investigated in great depth in the twentieth century, in particular the local theory, and the theory of solvable groups and nilpotent groups. It is too much to hope for a complete theory: the complexity becomes overwhelming when the group is large.
Finite groups are directly relevant to symmetry, when that is restricted to a finite number of transformations. It turns out that continuous symmetry, as modelled by Lie groups, also leads to finite groups, the Weyl groups. In this way, finite groups and their properties can enter centrally in questions, for example in theoretical physics, where their role is not initially obvious.
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