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:

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

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

3. \(0+x = x = x + 0\)

4. 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.