answersLogoWhite

0

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.

My source is linked below.

User Avatar

Wiki User

14y ago

Still curious? Ask our experts.

Chat with our AI personalities

ProfessorProfessor
I will give you the most educated answer.
Chat with Professor
TaigaTaiga
Every great hero faces trials, and you—yes, YOU—are no exception!
Chat with Taiga
FranFran
I've made my fair share of mistakes, and if I can help you avoid a few, I'd sure like to try.
Chat with Fran

Add your answer:

Earn +20 pts
Q: In group theory what is a group generator?
Write your answer...
Submit
Still have questions?
magnify glass
imp