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 $x{x}^{-1}=x^{-1}x=e$.
  • Associativity: $x\left(yz\right)=\left(xy\right)z$ 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 $(Z,+)$, the integers under addition. To see that it satisfies the group axioms, note that:

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