A group, $G$, is a [set](Sets.md) equipped with a [binary operation](Binary%20operations.md) referred to as the _group operation_ that combines any two elements in that group to form a third element in such a way that four conditions called _[group axioms](Group%20axioms.md)_ are satisfied. ^15feff
These conditions are:
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An elementary example of a group is the [integer group.](Groups.md#The%20integer%20group)^[Note that the rules for integer groups are all already taught in the earliest mathematics courses taken in childhood without invoking the same language as we do here. Defining the notion of a group and using this language is what allows us to apply and examine similar rules for other sets of numbers as well as mathematical objects.]
# Trivial groups
Since [groups](Groups.md) require the [inclusion of an identity element,](Groups.md#^3c624f) a group with the fewest possible elements is one that only contains an identity element and is denoted as the [single element set](Sets.md#Singletons) $G=\{ e \}$ where here we denote its identity element as $e.$
The remaining three group axioms also hold since there's only one possible function $G\times G \rightarrow G$ defined by a [binary operation](Binary%20operations.md) $\bullet$ such that $e\bullet e =e.$
Trivial groups are also all [Abelian](Abelian%20groups.md) since any function $e\bullet e$ also outputs the input element $e$.
%%see page 41 of Algebra chapter 0. It's also claimed that every singleton is a trivial group. Examine this and explain it.%%
# [Subgroups](Subgroups.md)
# Examples of groups
Less elementary examples of groups are indexed below.
* [GL(n;F)](GL(n;F).md)
* [[Lie groups]]s
* The [[Permutation group]]
* [[Number field]]s %%wait, check that number fields are groups%%
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# Proofs and Examples
## The integer group
An elementary example of a group goes as follows:
### Proof that the integer group is a group
## The integer group as a subgroup
#MathematicalFoundations/Algebra/AbstractAlgebra/GroupTheory
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