In abstract algebra, a generating set of a group is a subset that is not contained in any proper subgroup of the group. Equivalently, a generating set of a group is a subset such that 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.
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 generates G; 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 <x> equals the entire group G. For finite groups, it is also equivalent to saying that x has order |G|.
Read more about Generating Set Of A Group: Finitely Generated Group, Free Group, Frattini Subgroup, Examples
Famous quotes containing the words set and/or group:
“I do not set my life at a pin’s fee,
And for my soul, what can it do to that,
Being a thing immortal as itself?”
—William Shakespeare (1564–1616)
“My routines come out of total unhappiness. My audiences are my group therapy.”
—Joan Rivers (b. 1935)