Tour:Generating set of a group

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

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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?

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 $S$ of a group $G$ is termed a generating set if it satisfies the following equivalent conditions:

• For any element $g \in G$, we can write:

$g = a_1a_2 \ldots a_n$

where for each $a_i$, either $a_i \in S$ or $a_i^{-1} \in S$ (here, the $a_i$s are not necessarily distinct). In the situation where $S$ is a symmetric subset (i.e. $a_i \in S \implies a_i^{-1} \in S$) we do not need to throw in inverses. This happens, for instance, when $S$ is a union of subgroups of $G$.

• If $H$ is a proper subgroup of $G$ (i.e. $H$ is a subgroup of $G$ that is not equal to the whole of $G$), then $H$ cannot contain $S$.

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 $S$ is a subset of a group $G$ such that every element of $G$ is a power of some element of $S$, then $S$ is a generating set for $G$.

Some examples in abelian groups

• In the group of integers under addition, the singleton set $\{ 1 \}$ is a generating set. This is because every integer can be written as a sum of $1$s or $-1$s.
• In the group of integers under addition, the two-element set $\{ 2,3 \}$ is a generating set. To see this, note that every integer can expressed as a sum of $1$s or $-1$s, and both $1$ and $-1$ can be expressed in terms of $2$ and $3$.
• In the group of rational numbers, the set of all unit fractions $1/n, n \in \mathbb{N}$, form a generating set.

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