What does it mean for a set to generate a group?

In abstract algebra, a generating set of a group is a subset of the group set such that every element of the group can be expressed as a combination (under the group operation) of finitely many elements of the subset and their inverses.

Is a basis a generating set?

Like for vector spaces, a basis of a module is a linearly independent subset that is also a generating set.

How do you determine if a set is a generating set?

If V is a set of vectors from R^n and Span S = V, then we say that S is a generating set for V or that S generates V.

Do all groups have a generating set?

Every group has a set of generators. Groups that have one generator are called cyclic. Not every group is cyclic.

What is generate in linear algebra?

if there is extra structure on the vector space. A general definition of generation is as follows: let ei be elements of some structured set A. A subset B⊂A is said to be the structure generated by the ei if it is the intersection of all substructures of A containing the ei.

What is the generator for Z?

We say the group of integers under addition Z has only two generators, namely 1 and -1. However, Z can also be generated by any set of ‘relatively prime’ integers. (Integers having gcd 1).

What is the difference between basis and bases?

Basis means a starting point, base or foundation for an argument or hypothesis when used as a noun. Bases means foundations or starting points, checkpoints when used as a noun. A good way to remember the difference is Bases is the plural of base. Out of the two words, ‘basis’ is the most common.

What does a basis?

Definition of basis 1 : the bottom of something considered as its foundation. 2 : the principal component of something Fruit juice constitutes the basis of jelly.

What is a generating set of a vector space?

A generator of a vector space is also known as a spanning set. Some sources refer to a generator for rather than generator of. The two terms mean the same. It can also be said that S generates V (over K).

What is minimal generating set?

Definition. A generating set of a group is termed minimal or irredundant if any proper subset of the generating set, generates a strictly smaller (i.e. proper) subgroup. In other words, no generator can be dropped from the generating set.

Is a generator always 2?

Cyclic Groups and Generators So all the group elements {0,1,2,3,4} in Z5 can also be generated by 2. That is to say, 2 is also a generator for the group Z5. Not every element in a group is a generator. For example, the identity element in a group will never be a generator.

What does generate mean in maths?

Mathematics. to trace (a figure) by the motion of a point, straight line, or curve. to act as base for all the elements of a given set: The number 2 generates the set 2, 4, 8, 16.

What is the generating set of a group?

The set of all elements of a group is a generating set for the group. It is also the largest possible generating set. The set of all non-identity elements of a group is a generating set for the group. If is a subset of a group such that every element of is a power of some element of , then is a generating set for .

What is the difference between generators and generating sets?

The elements of the generating set are termed generators (the term is best used collectively for the generating set, rather than for the elements in isolation). A subset of a group is termed a generating set if it satisfies the following equivalent conditions: where for each , either or (here, the s are not necessarily distinct).

What is the generating set of the additive group?

The additive group of integers has 1 as a generating set. The element 2 is not a generating set, as the odd numbers will be missing. The two-element subset {3, 5} is a generating set, since (−5) + 3 + 3 = 1 (in fact, any pair of coprime numbers is, as a consequence of Bézout’s identity ).

How do you generate subsets of the same group?

Different subsets of the same group can be generating subsets. For example, if p and q are integers with gcd (p, q) = 1, then {p, q} also generates the group of integers under addition by Bézout’s identity .