In abstract algebra, a generating set of a group Gis a subset S such that every element of G can be expressed as the product of finitely many elements of S and their inverses.
More generally, if S is a subset of a group G, then <S>, the subgroup generated by S, is the smallest subgroup of G containing every element of S, meaning the intersection over all subgroups containing the elements of S; equivalently, is the subgroup of all elements of G that can be expressed as the finite product of elements in S and their inverses.
If G = , then we say S generatesG; and the elements in S are called generators or group generators. If S is the empty set, then <S> is the trivial group {e}, since we consider the empty product to be the identity.
When there is only a single element x in S, <S> is usually written as <x>. In this case, <x> is the cyclic subgroup of the powers of x, a cyclic group, and we say this group is generated by x. Equivalent to saying an element x generates a group is saying that it has order |G|, or that <x> equals the entire group G.
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In group theory, an alternating group is a group of even permutations of a finite set.
In abstract algebra, a generating set of a group is a subset of that group. In that subset, every element of the group can be expressed as the combination (under the group operation) of finitely many elements of the subset and their inverses.
A cyclic group, by definition, has only one generator. An example of an infinite cyclic group is the integers with addition. This group is generated by 1.
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Evariste Galois lived from 1811 till 1832. He died in a duel in Mary of 1832. He did not study mathematics at all until 1827 and appears to have concentrated on group theory in 1832.