A group $G$ is a pair $(X,\bullet )$ where $X$ is a set and $\bullet$ is a binary operation (of type $X\times X\to X$) subject to the four group axioms. Groups are algebraic structures. We write $x\bullet y$ for the application of $\bullet$ to $x,y\in X$, which must be defined. It is common to abbreviate $x\bullet y$ as $xy$ when $\bullet$ can be inferred from context. The group axioms (which govern the behavior of $\bullet$) are as follows.

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

1) The axiom of closure says that $x\bullet y\in X$, i.e. that combining two elements of $X$ using $\bullet$ yields another element of $X$. In other words, $X$ is closed under $\bullet$.

2) The axiom of associativity says that $\bullet$ is an associative operation, which justifies omitting parenthesis when describing the application of $\bullet$ to multiple arguments: We can write $\left(ab\right)\left(cd\right)$ and $a\left(b\right(cd\left)\right)$ and so on as simply $abcd$, because the order of application doesn't make a difference.

3) The axiom of identity says that there is some element $e$ in $X$ that $\bullet$ treats like an element that says "do nothing": 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 $ze=e=ez=z.$ 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 $0_G$.)

4) 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.

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. They seem to sit at a sweet spot with axioms that are restrictive enough to make groups easy to reason about and work with, but permissive enough that relevant group structure arises in many domains of math and physics. 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.