In mathematics, a permutation group is a group G whose elements are permutations of a given set M, and whose group operation is the composition of permutations in G (which are thought of as bijective functions from the set M to itself); the relationship is often written as (G,M). Note that the group of all permutations of a set is the symmetric group; the term permutation group is usually restricted to mean a subgroup of the symmetric group.
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Simple group
In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, a normal subgroup and the quotient group, and the process can be repeated. If the group is finite, then eventually one arrives at uniquely determined simple groups by the Jordan¿Hölder theorem.
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Order (group theory)
In group theory, a branch of mathematics, the term order is used in two closely related senses: The order of a group is its cardinality, i.e. , the number of elements in its set. The order, sometimes period, of an element a of a group is the smallest positive integer m such that a = e (where e denotes the identity element of the group, and a denotes the product of m copies of a). If no such m exists, a is said to have infinite order. All elements of finite groups have finite order.
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