# Group

A group is an abstraction of a collection of symmetries of an object. The collection of symmetries of a triangle (rotating by \(120^\circ\) or \(240^\circ\) degrees and flipping), rearrangements of a collection of objects (permutations), or rotations of a sphere, are all examples of groups. A group abstracts from these examples by forgetting what the symmetries are symmetries of, and only considers how symmetries behave.

A group \(G\) is a pair \((X, \bullet)\) where:

\(X\) is a set, called the “underlying set.” By abuse of notation, \(X\) is usually denoted by the same symbol as the group \(G\), which we will do for the rest of the article.

\(\bullet : G \times G \to G\) is a binary operation. That is, a function that takes two elements of a set and returns a third. We will abbreviate \(x \bullet y\) by \(xy\) when not ambiguous. This operation is subject to the following axioms:

**Closure:**\(\bullet\) is a function. For all \(x, y\) in \(X\), \(x \bullet y\) is defined and in \(X\). We abbreviate \(x \bullet y\) as \(xy\).**Identity:**There is an element \(e\) such that \(xe=ex=x\) for all \(x \in X\).**Inverses:**For each \(x\) in \(X\), there is an element \(x^{-1} \in X\) such that \(xx^{-1}=x^{-1}x=e\).**Associativity:**\(x(yz) = (xy)z\) for all \(x, y, z \in X\).

1) The set X is the collection of abstract symmetries that this group represents. “Abstract,” because these elements aren’t necessarily symmetries *of* something, but almost all examples will be.

2) The operation \(\bullet\) is the abstract composition operation.

3) The axiom of closure is redundant, since \(\bullet\) is defined as a function \(G \times G \to G\), but it is useful to emphasize this, as sometimes one can forget to check that a given subsets of symmetries of an object is closed under composition.

4) The axiom of identity says that there is an element \(e\) in \(G\) that is a do-nothing symmetry: If you apply \(\bullet\) to \(e\) and \(x\), then \(\bullet\) simply returns \(x\). The identity is unique: Given two elements \(e\) and \(z\) that satisfy axiom 2, we have \(ze = ez = z.\) Thus, we can speak of “the identity” \(e\) of \(G\). This justifies the use of \(e\) in the axiom of inversion: axioms 1 through 3 ensure that \(e\) exists and is unique, so we can reference it in axiom 4.

\(e\) is often written \(1\) or \(1_G\), because \(\bullet\) is often treated as an analog of multiplication on the set \(X\), and \(1\) is the multiplicative identity. (Sometimes, e.g. in the case of rings, \(\bullet\) is treated as an analog of addition, in which case the identity is often written \(0\) or \(0_G\).)

5) The axiom of inverses says that for every element \(x\) in \(X\), there is some other element \(y\) that \(\bullet\) treats like the opposite of \(x\), in the sense that \(xy = e\) and vice versa. The inverse of \(x\) is usually written \(x^{-1}\), or sometimes \((-x)\) in cases where \(\bullet\) is analogous to addition.

6) The axiom of associativity says that \bullet behaves like composition of functions. When composing a bunch of functions, it doesn’t matter what order the individual compositions are computed in. When composing \(f\), \(g\), and \(h\), we can compute \(g \circ f\), and then compute \(h \circ (g \circ f)\), or we can compute \(h \circ g\) and then compute \((h \circ g) \circ f\), and we will get the same result.

# Examples

The most familiar example of a group is perhaps \((\mathbb{Z}, +)\), the integers under addition. To see that it satisfies the group axioms, note that:

(a) \(\mathbb{Z}\) is a set, and (b) \(+\) is a function of type \(\mathbb Z \times \mathbb Z \to \mathbb Z\)

\((x+y)+z=x+(y+z)\)

\(0+x = x = x + 0\)

Every element \(x\) has an inverse \(-x\), because \(x + (-x) = 0\).

For more examples, see the examples page.

# Notation

Given a group \(G = (X, \bullet)\), we say ”\(X\) forms a group under \(\bullet\).” \(X\) is called the underlying set of \(G\), and \(\bullet\) is called the *group operation*.

\(x \bullet y\) is usually abbreviated \(xy\).

\(G\) is generally allowed to substitute for \(X\) when discussing the group. For example, we say that the elements \(x, y \in X\) are “in \(G\),” and sometimes write “\(x, y \in G\)” or talk about the “elements of \(G\).”

The order of a group, written \(|G|\), is the size \(|X|\) of its underlying set: If \(X\) has nine elements, then \(|G|=9\) and we say that \(G\) has order nine.

# Resources

Groups are a ubiquitous and useful algebraic structure. Whenever it makes sense to talk about symmetries of a mathematical object, or physical system, groups pop up. For a discussion of group theory and its various applications, refer to the group theory page.

A group is a monoid with inverses, and an associative loop. For more on how groups relate to other algebraic structures, refer to the tree of algebraic structures.

Children:

- Order of a group
- Abelian group
A group where the operation commutes. Named after Niels Henrik Abel.

- Group: Examples
Why would anyone have invented groups, anyway? What were the historically motivating examples, and what examples are important today?

- Group: Exercises
Test your understanding of the definition of a group with these exercises.

- Group homomorphism
A group homomorphism is a “function between groups” that “respects the group structure”.

- Cyclic group
Cyclic groups form one of the most simple classes of groups.

- Symmetric group
The symmetric groups form the fundamental link between group theory and the notion of symmetry.

- Group isomorphism
“Isomorphism” is the proper notion of “sameness” or “equality” among groups.

- Order of a group element
- Dihedral group
The dihedral groups are natural examples of groups, arising from the symmetries of regular polygons.

- Group conjugate
Conjugation lets us perform permutations “from the point of view of” another permutation.

- Normal subgroup
Normal subgroups are subgroups which are in some sense “the same from all points of view”.

- Alternating group
The alternating group is the only normal subgroup of the symmetric group (on five or more generators).

- Group coset
- Simple group
The simple groups form the “building blocks” of group theory, analogously to the prime numbers in number theory.

- Prime order groups are cyclic
This is the first step on the road to classifying the finite groups.

- Lagrange theorem on subgroup size
Lagrange’s Theorem is an important restriction on the sizes of subgroups of a finite group.

- Cauchy's theorem on subgroup existence
Cauchy’s theorem is a useful condition for the existence of cyclic subgroups of finite groups.

- Group orbit
- Cyclic Group Intro (Math 0)
A finite cyclic group is a little bit like a clock.

- Subgroup
A group that lives inside a bigger group.

- Group presentation
Presentations are a fairly compact way of expressing groups.

- Every group is a quotient of a free group
- Free group
The free group is “the purest way to make a group containing a given set”.

Parents:

- Group theory
What kinds of symmetry can an object have?

- Algebraic structure

Some of the aliases and stuff around groups have been using the term “algebraic group”; this has a different technical meaning in mathematics and should be avoided.