# Tour:Generating set of a group

From Groupprops

**This article adapts material from the main article:** generating set of a group

This page is part of the Groupprops guided tour for beginners (Jump to beginning of tour)PREVIOUS: Lagrange's theorem|UP: Introduction three (beginners)|NEXT: Subgroup generated by a subset

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WHAT YOU NEED TO DO:

- Understand the two definitions of generating set of a group.
- Convince yourself that the two definitions are equivalent.
PONDER: To what extent is a group described by its generating set? What do you think are the possible ways in which generating sets may be important?

## Contents

## Definition

### Symbol-free definition

A subset of a group is termed a **generating set** if it satisfies the following equivalent conditions:

- Every element of the group can be expressed as a word in terms of the elements of this subset, i.e., it can be expressed using the elements of the subset by means of the group operations of multiplication and inversion. (note that if the subset is a symmetric subset, i.e., it contains the identity and is closed under taking inverses, then every element of the group must be a product of elements in the subset. Symmetric subsets arise, for instance, when we take a union of subgroups).
- There is no proper subgroup of the group containing this subset

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).

### Definition with symbols

A subset of a group is termed a **generating set** if it satisfies the following equivalent conditions:

- For any element , we can write:

where for each , either or (here, the s are not necessarily distinct). In the situation where is a symmetric subset (i.e. ) we do not need to throw in inverses. This happens, for instance, when is a union of subgroups of .

- If is a proper subgroup of (i.e. is a subgroup of that is not equal to the whole of ), then cannot contain .

## Examples

### Extreme examples

- 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 .

### Some examples in abelian groups

- In the group of integers under addition, the singleton set is a generating set. This is because every integer can be written as a sum of s or s.
- In the group of integers under addition, the two-element set is a generating set. To see this, note that every integer can expressed as a sum of s or s, and both and can be expressed in terms of and .
- In the group of rational numbers, the set of all unit fractions , form a generating set.

This page is part of the Groupprops guided tour for beginners. Make notes of any doubts, confusions or comments you have about this page before proceeding.PREVIOUS: Lagrange's theorem|UP: Introduction three (beginners)|NEXT: Subgroup generated by a subset