Group (mathematics)

One of the more familiar groups is the set of integers

$${\displaystyle \mathbb {Z} =\{\ldots ,-4,-3,-2,-1,0,1,2,3,4,\ldots \}}$$

together with addition.[3] For any two integers ${\displaystyle a}$ and ${\displaystyle b}$, the sum ${\displaystyle a+b}$ is also an integer; this closure property says that ${\displaystyle +}$ is a binary operation on ${\displaystyle \mathbb {Z} }$. The following properties of integer addition serve as a model for the group axioms in the definition below.

• For all integers ${\displaystyle a}$, ${\displaystyle b}$ and ${\displaystyle c}$, one has ${\displaystyle (a+b)+c=a+(b+c)}$. Expressed in words, adding ${\displaystyle a}$ to ${\displaystyle b}$ first, and then adding the result to ${\displaystyle c}$ gives the same final result as adding ${\displaystyle a}$ to the sum of ${\displaystyle b}$ and ${\displaystyle c}$. This property is known as associativity.
• If ${\displaystyle a}$ is any integer, then ${\displaystyle 0+a=a}$ and ${\displaystyle a+0=a}$. Zero is called the identity element of addition because adding it to any integer returns the same integer.
• For every integer ${\displaystyle a}$, there is an integer ${\displaystyle b}$ such that ${\displaystyle a+b=0}$ and ${\displaystyle b+a=0}$. The integer ${\displaystyle b}$ is called the inverse element of the integer ${\displaystyle a}$ and is denoted ${\displaystyle -a}$.

The integers, together with the operation ${\displaystyle +}$, form a mathematical object belonging to a broad class sharing similar structural aspects. To appropriately understand these structures as a collective, the following definition is developed.

A group is a non-empty set ${\displaystyle G}$ together with a binary operation on ${\displaystyle G}$, here denoted "${\displaystyle \cdot }$", that combines any two elements ${\displaystyle a}$ and ${\displaystyle b}$ of ${\displaystyle G}$ to form an element of ${\displaystyle G}$, denoted ${\displaystyle a\cdot b}$, such that the following three requirements, known as group axioms, are satisfied:[5][6][7][lower-alpha 1]

Associativity

For all ${\displaystyle a}$, ${\displaystyle b}$, ${\displaystyle c}$ in ${\displaystyle G}$, one has ${\displaystyle (a\cdot b)\cdot c=a\cdot (b\cdot c)}$.

Identity element

There exists an element ${\displaystyle e}$ in ${\displaystyle G}$ such that, for every ${\displaystyle a}$ in ${\displaystyle G}$, one has ${\displaystyle e\cdot a=a}$ and ${\displaystyle a\cdot e=a}$.

Such an element is unique (see below). It is called the identity element of the group.

Inverse element

For each ${\displaystyle a}$ in ${\displaystyle G}$, there exists an element ${\displaystyle b}$ in ${\displaystyle G}$ such that ${\displaystyle a\cdot b=e}$ and ${\displaystyle b\cdot a=e}$, where ${\displaystyle e}$ is the identity element.

For each ${\displaystyle a}$, the element ${\displaystyle b}$ is unique (see below); it is called the inverse of ${\displaystyle a}$ and is commonly denoted ${\displaystyle a^{-1}}$.

Formally, the group is the ordered pair of a set and a binary operation on this set that satisfies the group axioms. The set is called the underlying set of the group, and the operation is called the group operation or the group law.

A group and its underlying set are thus two different mathematical objects. To avoid cumbersome notation, it is common to abuse notation by using the same symbol to denote both. This reflects also an informal way of thinking: that the group is the same as the set except that it has been enriched by additional structure provided by the operation.

For example, consider the set of real numbers ${\displaystyle \mathbb {R} }$, which has the operations of addition ${\displaystyle a+b}$ and multiplication ${\displaystyle ab}$. Formally, ${\displaystyle \mathbb {R} }$ is a set, ${\displaystyle (\mathbb {R} ,+)}$ is a group, and ${\displaystyle (\mathbb {R} ,+,\cdot )}$ is a field. But it is common to write ${\displaystyle \mathbb {R} }$ to denote any of these three objects.

The additive group of the field ${\displaystyle \mathbb {R} }$ is the group whose underlying set is ${\displaystyle \mathbb {R} }$ and whose operation is addition. The multiplicative group of the field ${\displaystyle \mathbb {R} }$ is the group ${\displaystyle \mathbb {R} ^{\times }}$ whose underlying set is the set of nonzero real numbers ${\displaystyle \mathbb {R} \smallsetminus \{0\}}$ and whose operation is multiplication.

More generally, one speaks of an additive group whenever the group operation is notated as addition; in this case, the identity is typically denoted ${\displaystyle 0}$, and the inverse of an element ${\displaystyle x}$ is denoted ${\displaystyle -x}$. Similarly, one speaks of a multiplicative group whenever the group operation is notated as multiplication; in this case, the identity is typically denoted ${\displaystyle 1}$, and the inverse of an element ${\displaystyle x}$ is denoted ${\displaystyle x^{-1}}$. In a multiplicative group, the operation symbol is usually omitted entirely, so that the operation is denoted by juxtaposition, ${\displaystyle ab}$ instead of ${\displaystyle a\cdot b}$.

The definition of a group does not require that ${\displaystyle a\cdot b=b\cdot a}$ for all elements ${\displaystyle a}$ and ${\displaystyle b}$ in ${\displaystyle G}$. If this additional condition holds, then the operation is said to be commutative, and the group is called an abelian group. It is a common convention that for an abelian group either additive or multiplicative notation may be used, but for a nonabelian group only multiplicative notation is used.

Several other notations are commonly used for groups whose elements are not numbers. For a group whose elements are functions, the operation is often function composition ${\displaystyle f\circ g}$; then the identity may be denoted id. In the more specific cases of geometric transformation groups, symmetry groups, permutation groups, and automorphism groups, the symbol ${\displaystyle \circ }$ is often omitted, as for multiplicative groups. Many other variants of notation may be encountered.

Two figures in the plane are congruent if one can be changed into the other using a combination of rotations, reflections, and translations. Any figure is congruent to itself. However, some figures are congruent to themselves in more than one way, and these extra congruences are called symmetries. A square has eight symmetries. These are:

The elements of the symmetry group of the square, ${\displaystyle \mathrm {D} _{4}}$. Vertices are identified by color or number.

${\displaystyle \mathrm {id} }$ (keeping it as it is)	${\displaystyle r_{1}}$ (rotation by 90° clockwise)	${\displaystyle r_{2}}$ (rotation by 180°)	${\displaystyle r_{3}}$ (rotation by 270° clockwise)
${\displaystyle f_{\mathrm {v} }}$ (vertical reflection)	${\displaystyle f_{\mathrm {h} }}$ (horizontal reflection)	${\displaystyle f_{\mathrm {d} }}$ (diagonal reflection)	${\displaystyle f_{\mathrm {c} }}$ (counter-diagonal reflection)

• the identity operation leaving everything unchanged, denoted id;
• rotations of the square around its center by 90°, 180°, and 270° clockwise, denoted by ${\displaystyle r_{1}}$, ${\displaystyle r_{2}}$ and ${\displaystyle r_{3}}$, respectively;
• reflections about the horizontal and vertical middle line (${\displaystyle f_{\mathrm {v} }}$ and ${\displaystyle f_{\mathrm {h} }}$), or through the two diagonals (${\displaystyle f_{\mathrm {d} }}$ and ${\displaystyle f_{\mathrm {c} }}$).

These symmetries are functions. Each sends a point in the square to the corresponding point under the symmetry. For example, ${\displaystyle r_{1}}$ sends a point to its rotation 90° clockwise around the square's center, and ${\displaystyle f_{\mathrm {h} }}$ sends a point to its reflection across the square's vertical middle line. Composing two of these symmetries gives another symmetry. These symmetries determine a group called the dihedral group of degree four, denoted ${\displaystyle \mathrm {D} _{4}}$. The underlying set of the group is the above set of symmetries, and the group operation is function composition.[8] Two symmetries are combined by composing them as functions, that is, applying the first one to the square, and the second one to the result of the first application. The result of performing first ${\displaystyle a}$ and then ${\displaystyle b}$ is written symbolically from right to left as ${\displaystyle b\circ a}$ ("apply the symmetry ${\displaystyle b}$ after performing the symmetry ${\displaystyle a}$"). This is the usual notation for composition of functions.

The group table lists the results of all such compositions possible. For example, rotating by 270° clockwise (${\displaystyle r_{3}}$) and then reflecting horizontally (${\displaystyle f_{\mathrm {h} }}$) is the same as performing a reflection along the diagonal (${\displaystyle f_{\mathrm {d} }}$). Using the above symbols, highlighted in blue in the group table:

$${\displaystyle f_{\mathrm {h} }\circ r_{3}=f_{\mathrm {d} }.}$$

Group table of ${\displaystyle \mathrm {D} _{4}}$

${\displaystyle \circ }$	${\displaystyle \mathrm {id} }$	${\displaystyle r_{1}}$	${\displaystyle r_{2}}$	${\displaystyle r_{3}}$	${\displaystyle f_{\mathrm {v} }}$	${\displaystyle f_{\mathrm {h} }}$	${\displaystyle f_{\mathrm {d} }}$	${\displaystyle f_{\mathrm {c} }}$
${\displaystyle \mathrm {id} }$	${\displaystyle \mathrm {id} }$	${\displaystyle r_{1}}$	${\displaystyle r_{2}}$	${\displaystyle r_{3}}$	${\displaystyle f_{\mathrm {v} }}$	${\displaystyle f_{\mathrm {h} }}$	${\displaystyle f_{\mathrm {d} }}$	${\displaystyle f_{\mathrm {c} }}$
${\displaystyle r_{1}}$	${\displaystyle r_{1}}$	${\displaystyle r_{2}}$	${\displaystyle r_{3}}$	${\displaystyle \mathrm {id} }$	${\displaystyle f_{\mathrm {c} }}$	${\displaystyle f_{\mathrm {d} }}$	${\displaystyle f_{\mathrm {v} }}$	${\displaystyle f_{\mathrm {h} }}$
${\displaystyle r_{2}}$	${\displaystyle r_{2}}$	${\displaystyle r_{3}}$	${\displaystyle \mathrm {id} }$	${\displaystyle r_{1}}$	${\displaystyle f_{\mathrm {h} }}$	${\displaystyle f_{\mathrm {v} }}$	${\displaystyle f_{\mathrm {c} }}$	${\displaystyle f_{\mathrm {d} }}$
${\displaystyle r_{3}}$	${\displaystyle r_{3}}$	${\displaystyle \mathrm {id} }$	${\displaystyle r_{1}}$	${\displaystyle r_{2}}$	${\displaystyle f_{\mathrm {d} }}$	${\displaystyle f_{\mathrm {c} }}$	${\displaystyle f_{\mathrm {h} }}$	${\displaystyle f_{\mathrm {v} }}$
${\displaystyle f_{\mathrm {v} }}$	${\displaystyle f_{\mathrm {v} }}$	${\displaystyle f_{\mathrm {d} }}$	${\displaystyle f_{\mathrm {h} }}$	${\displaystyle f_{\mathrm {c} }}$	${\displaystyle \mathrm {id} }$	${\displaystyle r_{2}}$	${\displaystyle r_{1}}$	${\displaystyle r_{3}}$
${\displaystyle f_{\mathrm {h} }}$	${\displaystyle f_{\mathrm {h} }}$	${\displaystyle f_{\mathrm {c} }}$	${\displaystyle f_{\mathrm {v} }}$	${\displaystyle f_{\mathrm {d} }}$	${\displaystyle r_{2}}$	${\displaystyle \mathrm {id} }$	${\displaystyle r_{3}}$	${\displaystyle r_{1}}$
${\displaystyle f_{\mathrm {d} }}$	${\displaystyle f_{\mathrm {d} }}$	${\displaystyle f_{\mathrm {h} }}$	${\displaystyle f_{\mathrm {c} }}$	${\displaystyle f_{\mathrm {v} }}$	${\displaystyle r_{3}}$	${\displaystyle r_{1}}$	${\displaystyle \mathrm {id} }$	${\displaystyle r_{2}}$
${\displaystyle f_{\mathrm {c} }}$	${\displaystyle f_{\mathrm {c} }}$	${\displaystyle f_{\mathrm {v} }}$	${\displaystyle f_{\mathrm {d} }}$	${\displaystyle f_{\mathrm {h} }}$	${\displaystyle r_{1}}$	${\displaystyle r_{3}}$	${\displaystyle r_{2}}$	${\displaystyle \mathrm {id} }$
The elements ${\displaystyle \mathrm {id} }$, ${\displaystyle r_{1}}$, ${\displaystyle r_{2}}$, and ${\displaystyle r_{3}}$ form a subgroup whose group table is highlighted in   red (upper left region). A left and right coset of this subgroup are highlighted in   green (in the last row) and   yellow (last column), respectively. The result of the composition ${\displaystyle f_{\mathrm {h} }\circ r_{3}}$, the symmetry ${\displaystyle f_{\mathrm {d} }}$, is highlighted in   blue (below table center).

Given this set of symmetries and the described operation, the group axioms can be understood as follows.

Binary operation: Composition is a binary operation. That is, ${\displaystyle a\circ b}$ is a symmetry for any two symmetries ${\displaystyle a}$ and ${\displaystyle b}$. For example,

$${\displaystyle r_{3}\circ f_{\mathrm {h} }=f_{\mathrm {c} },}$$

that is, rotating 270° clockwise after reflecting horizontally equals reflecting along the counter-diagonal (${\displaystyle f_{\mathrm {c} }}$). Indeed, every other combination of two symmetries still gives a symmetry, as can be checked using the group table.

Associativity: The associativity axiom deals with composing more than two symmetries: Starting with three elements ${\displaystyle a}$, ${\displaystyle b}$ and ${\displaystyle c}$ of ${\displaystyle \mathrm {D} _{4}}$, there are two possible ways of using these three symmetries in this order to determine a symmetry of the square. One of these ways is to first compose ${\displaystyle a}$ and ${\displaystyle b}$ into a single symmetry, then to compose that symmetry with ${\displaystyle c}$. The other way is to first compose ${\displaystyle b}$ and ${\displaystyle c}$, then to compose the resulting symmetry with ${\displaystyle a}$. These two ways must give always the same result, that is,

$${\displaystyle (a\circ b)\circ c=a\circ (b\circ c),}$$

For example, ${\displaystyle (f_{\mathrm {d} }\circ f_{\mathrm {v} })\circ r_{2}=f_{\mathrm {d} }\circ (f_{\mathrm {v} }\circ r_{2})}$ can be checked using the group table:

$${\displaystyle {\begin{aligned}(f_{\mathrm {d} }\circ f_{\mathrm {v} })\circ r_{2}&=r_{3}\circ r_{2}=r_{1}\\f_{\mathrm {d} }\circ (f_{\mathrm {v} }\circ r_{2})&=f_{\mathrm {d} }\circ f_{\mathrm {h} }=r_{1}.\end{aligned}}}$$

Identity element: The identity element is ${\displaystyle \mathrm {id} }$, as it does not change any symmetry ${\displaystyle a}$ when composed with it either on the left or on the right.

Inverse element: Each symmetry has an inverse: ${\displaystyle \mathrm {id} }$, the reflections ${\displaystyle f_{\mathrm {h} }}$, ${\displaystyle f_{\mathrm {v} }}$, ${\displaystyle f_{\mathrm {d} }}$, ${\displaystyle f_{\mathrm {c} }}$ and the 180° rotation ${\displaystyle r_{2}}$ are their own inverse, because performing them twice brings the square back to its original orientation. The rotations ${\displaystyle r_{3}}$ and ${\displaystyle r_{1}}$ are each other's inverses, because rotating 90° and then rotation 270° (or vice versa) yields a rotation over 360° which leaves the square unchanged. This is easily verified on the table.

In contrast to the group of integers above, where the order of the operation is immaterial, it does matter in ${\displaystyle \mathrm {D} _{4}}$, as, for example, ${\displaystyle f_{\mathrm {h} }\circ r_{1}=f_{\mathrm {c} }}$ but ${\displaystyle r_{1}\circ f_{\mathrm {h} }=f_{\mathrm {d} }}$. In other words, ${\displaystyle \mathrm {D} _{4}}$ is not abelian.

The group axioms imply that the identity element is unique: If ${\displaystyle e}$ and ${\displaystyle f}$ are identity elements of a group, then ${\displaystyle e=e\cdot f=f}$. Therefore, it is customary to speak of the identity.[27]

The group axioms also imply that the inverse of each element is unique: if a group element ${\displaystyle a}$ has both ${\displaystyle b}$ and ${\displaystyle c}$ as inverses, then

${\displaystyle {\begin{aligned}b&=b\cdot e&&{\text{(}}e{\text{ is the identity element)}}\\&=b\cdot (a\cdot c)&&{\text{(}}c{\text{ is an inverse)}}\\&=(b\cdot a)\cdot c&&{\text{(associativity)}}\\&=e\cdot c&&{\text{(}}b{\text{ is an inverse)}}\\&=c&&{\text{(}}e{\text{ is the identity element)}}\end{aligned}}}$

Therefore, it is customary to speak of the inverse of an element.[27]

Given elements ${\displaystyle a}$ and ${\displaystyle b}$ of a group ${\displaystyle G}$, there is a unique solution ${\displaystyle x}$ in ${\displaystyle G}$ to the equation ${\displaystyle a\cdot x=b}$, namely ${\displaystyle a^{-1}\cdot b}$.[lower-alpha 3][28] It follows that for each ${\displaystyle a}$ in ${\displaystyle G}$, the function ${\displaystyle G\to G}$ that maps each ${\displaystyle x}$ to ${\displaystyle a\cdot x}$ is a bijection; it is called left multiplication by ${\displaystyle a}$ or left translation by ${\displaystyle a.}$

Similarly, given ${\displaystyle a}$ and ${\displaystyle b}$, the unique solution to ${\displaystyle x\cdot a=b}$ is ${\displaystyle b\cdot a^{-1}}$. For each ${\displaystyle a}$, the function ${\displaystyle G\to G}$ that maps each ${\displaystyle x}$ to ${\displaystyle x\cdot a}$ is a bijection called right multiplication by ${\displaystyle a}$ or right translation by ${\displaystyle a.}$

The group axioms for identity and inverses may be "weakened" to assert only the existence of a left identity and left inverses. From these one-sided axioms, one can prove that the left identity is also a right identity and a left inverse is also a right inverse for the same element. Since they define exactly the same structures as groups, collectively the axioms are not weaker.[29]

In particular, assuming associativity and the existence of a left identity ${\displaystyle e}$ (that is, ${\displaystyle e\cdot f=f}$) and a left inverse ${\displaystyle f^{-1}}$ for each element ${\displaystyle f}$ (that is, ${\displaystyle f^{-1}\cdot f=e}$), one can show that every left inverse is also a right inverse of the same element as follows.[29] Indeed, one has

${\displaystyle {\begin{aligned}f\cdot f^{-1}&=e\cdot (f\cdot f^{-1})&&{\text{(left identity)}}\\&=((f^{-1})^{-1}\cdot f^{-1})\cdot (f\cdot f^{-1})&&{\text{(left inverse)}}\\&=(f^{-1})^{-1}\cdot ((f^{-1}\cdot f)\cdot f^{-1})&&{\text{(associativity)}}\\&=(f^{-1})^{-1}\cdot (e\cdot f^{-1})&&{\text{(left inverse)}}\\&=(f^{-1})^{-1}\cdot f^{-1}&&{\text{(left identity)}}\\&=e&&{\text{(left inverse)}}\end{aligned}}}$

Similarly, the left identity is also a right identity:[29]

${\displaystyle {\begin{aligned}f\cdot e&=f\cdot (f^{-1}\cdot f)&&{\text{(left inverse)}}\\&=(f\cdot f^{-1})\cdot f&&{\text{(associativity)}}\\&=e\cdot f&&{\text{(right inverse)}}\\&=f&&{\text{(left identity)}}\end{aligned}}}$

These proofs require all three axioms (associativity, existence of left identity and existence of left inverse). For a structure with a looser definition (like a semigroup) one may have, for example, that a left identity is not necessarily a right identity.

The same result can be obtained by only assuming the existence of a right identity and a right inverse.

However, only assuming the existence of a left identity and a right inverse (or vice versa) is not sufficient to define a group. For example, consider the set ${\displaystyle G=\{e,f\}}$ with the operator ${\displaystyle \cdot }$ satisfying ${\displaystyle e\cdot e=f\cdot e=e}$ and ${\displaystyle e\cdot f=f\cdot f=f}$. This structure does have a left identity (namely, ${\displaystyle e}$), and each element has a right inverse (which is ${\displaystyle e}$ for both elements). Furthermore, this operation is associative (since the product of any number of elements is always equal to the rightmost element in that product, regardless of the order in which these operations are done). However, ${\displaystyle (G,\cdot )}$ is not a group, since it lacks a right identity.

Group homomorphisms[lower-alpha 5] are functions that respect group structure; they may be used to relate two groups. A homomorphism from a group ${\displaystyle (G,\cdot )}$ to a group ${\displaystyle (H,*)}$ is a function ${\displaystyle \varphi \colon G\to H}$ such that

${\displaystyle \varphi (a\cdot b)=\varphi (a)*\varphi (b)}$ for all elements ${\displaystyle a}$ and ${\displaystyle b}$ in ${\displaystyle G}$.

It would be natural to require also that ${\displaystyle \varphi }$ respect identities, ${\displaystyle \varphi (1_{G})=1_{H}}$, and inverses, ${\displaystyle \varphi (a^{-1})=\varphi (a)^{-1}}$ for all ${\displaystyle a}$ in ${\displaystyle G}$. However, these additional requirements need not be included in the definition of homomorphisms, because they are already implied by the requirement of respecting the group operation.[30]

The identity homomorphism of a group ${\displaystyle G}$ is the homomorphism ${\displaystyle \iota _{G}\colon G\to G}$ that maps each element of ${\displaystyle G}$ to itself. An inverse homomorphism of a homomorphism ${\displaystyle \varphi \colon G\to H}$ is a homomorphism ${\displaystyle \psi \colon H\to G}$ such that ${\displaystyle \psi \circ \varphi =\iota _{G}}$ and ${\displaystyle \varphi \circ \psi =\iota _{H}}$, that is, such that ${\displaystyle \psi {\bigl (}\varphi (g){\bigr )}=g}$ for all ${\displaystyle g}$ in ${\displaystyle G}$ and such that ${\displaystyle \varphi {\bigl (}\psi (h){\bigr )}=h}$ for all ${\displaystyle h}$ in ${\displaystyle H}$. An isomorphism is a homomorphism that has an inverse homomorphism; equivalently, it is a bijective homomorphism. Groups ${\displaystyle G}$ and ${\displaystyle H}$ are called isomorphic if there exists an isomorphism ${\displaystyle \varphi \colon G\to H}$. In this case, ${\displaystyle H}$ can be obtained from ${\displaystyle G}$ simply by renaming its elements according to the function ${\displaystyle \varphi }$; then any statement true for ${\displaystyle G}$ is true for ${\displaystyle H}$, provided that any specific elements mentioned in the statement are also renamed.

The collection of all groups, together with the homomorphisms between them, form a category, the category of groups.[31]

An injective homomorphism ${\displaystyle \phi \colon G'\to G}$ factors canonically as an isomorphism followed by an inclusion, ${\displaystyle G'\;{\stackrel {\sim }{\to }}\;H\hookrightarrow G}$ for some subgroup H of G. Injective homomorphisms are the monomorphisms in the category of groups.

Informally, a subgroup is a group ${\displaystyle H}$ contained within a bigger one, ${\displaystyle G}$: it has a subset of the elements of ${\displaystyle G}$, with the same operation.[32] Concretely, this means that the identity element of ${\displaystyle G}$ must be contained in ${\displaystyle H}$, and whenever ${\displaystyle h_{1}}$ and ${\displaystyle h_{2}}$ are both in ${\displaystyle H}$, then so are ${\displaystyle h_{1}\cdot h_{2}}$ and ${\displaystyle h_{1}^{-1}}$, so the elements of ${\displaystyle H}$, equipped with the group operation on ${\displaystyle G}$ restricted to ${\displaystyle H}$, indeed form a group. In this case, the inclusion map ${\displaystyle H\to G}$ is a homomorphism.

In the example of symmetries of a square, the identity and the rotations constitute a subgroup ${\displaystyle R=\{\mathrm {id} ,r_{1},r_{2},r_{3}\}}$, highlighted in red in the group table of the example: any two rotations composed are still a rotation, and a rotation can be undone by (i.e., is inverse to) the complementary rotations 270° for 90°, 180° for 180°, and 90° for 270°. The subgroup test provides a necessary and sufficient condition for a nonempty subset H of a group G to be a subgroup: it is sufficient to check that ${\displaystyle g^{-1}\cdot h\in H}$ for all elements ${\displaystyle g}$ and ${\displaystyle h}$ in ${\displaystyle H}$. Knowing a group's subgroups is important in understanding the group as a whole.[lower-alpha 6]

Given any subset ${\displaystyle S}$ of a group ${\displaystyle G}$, the subgroup generated by ${\displaystyle S}$ consists of all products of elements of ${\displaystyle S}$ and their inverses. It is the smallest subgroup of ${\displaystyle G}$ containing ${\displaystyle S}$.[33] In the example of symmetries of a square, the subgroup generated by ${\displaystyle r_{2}}$ and ${\displaystyle f_{\mathrm {v} }}$ consists of these two elements, the identity element ${\displaystyle \mathrm {id} }$, and the element ${\displaystyle f_{\mathrm {h} }=f_{\mathrm {v} }\cdot r_{2}}$. Again, this is a subgroup, because combining any two of these four elements or their inverses (which are, in this particular case, these same elements) yields an element of this subgroup.

In many situations it is desirable to consider two group elements the same if they differ by an element of a given subgroup. For example, in the symmetry group of a square, once any reflection is performed, rotations alone cannot return the square to its original position, so one can think of the reflected positions of the square as all being equivalent to each other, and as inequivalent to the unreflected positions; the rotation operations are irrelevant to the question whether a reflection has been performed. Cosets are used to formalize this insight: a subgroup ${\displaystyle H}$ determines left and right cosets, which can be thought of as translations of ${\displaystyle H}$ by an arbitrary group element ${\displaystyle g}$. In symbolic terms, the left and right cosets of ${\displaystyle H}$, containing an element ${\displaystyle g}$, are

${\displaystyle gH=\{g\cdot h\mid h\in H\}}$ and ${\displaystyle Hg=\{h\cdot g\mid h\in H\}}$, respectively.[34]

The left cosets of any subgroup ${\displaystyle H}$ form a partition of ${\displaystyle G}$; that is, the union of all left cosets is equal to ${\displaystyle G}$ and two left cosets are either equal or have an empty intersection.[35] The first case ${\displaystyle g_{1}H=g_{2}H}$ happens precisely when ${\displaystyle g_{1}^{-1}\cdot g_{2}\in H}$, i.e., when the two elements differ by an element of ${\displaystyle H}$. Similar considerations apply to the right cosets of ${\displaystyle H}$. The left cosets of ${\displaystyle H}$ may or may not be the same as its right cosets. If they are (that is, if all ${\displaystyle g}$ in ${\displaystyle G}$ satisfy ${\displaystyle gH=Hg}$), then ${\displaystyle H}$ is said to be a normal subgroup.

In ${\displaystyle \mathrm {D} _{4}}$, the group of symmetries of a square, with its subgroup ${\displaystyle R}$ of rotations, the left cosets ${\displaystyle gR}$ are either equal to ${\displaystyle R}$, if ${\displaystyle g}$ is an element of ${\displaystyle R}$ itself, or otherwise equal to ${\displaystyle U=f_{\mathrm {c} }R=\{f_{\mathrm {c} },f_{\mathrm {d} },f_{\mathrm {v} },f_{\mathrm {h} }\}}$ (highlighted in green in the group table of ${\displaystyle \mathrm {D} _{4}}$). The subgroup ${\displaystyle R}$ is normal, because ${\displaystyle f_{\mathrm {c} }R=U=Rf_{\mathrm {c} }}$ and similarly for the other elements of the group. (In fact, in the case of ${\displaystyle \mathrm {D} _{4}}$, the cosets generated by reflections are all equal: ${\displaystyle f_{\mathrm {h} }R=f_{\mathrm {v} }R=f_{\mathrm {d} }R=f_{\mathrm {c} }R}$.)

Suppose that ${\displaystyle N}$ is a normal subgroup of a group ${\displaystyle G}$, and

$${\displaystyle G/N=\{gN\mid g\in G\}}$$

denotes its set of cosets. Then there is a unique group law on ${\displaystyle G/N}$ for which the map ${\displaystyle G\to G/N}$ sending each element ${\displaystyle g}$ to ${\displaystyle gN}$ is a homomorphism. Explicitly, the product of two cosets ${\displaystyle gN}$ and ${\displaystyle hN}$ is ${\displaystyle (gh)N}$, the coset ${\displaystyle eN=N}$ serves as the identity of ${\displaystyle G/N}$, and the inverse of ${\displaystyle gN}$ in the quotient group is ${\displaystyle (gN)^{-1}=\left(g^{-1}\right)N}$. The group ${\displaystyle G/N}$, read as "${\displaystyle G}$ modulo ${\displaystyle N}$",[36] is called a quotient group or factor group. The quotient group can alternatively be characterized by a universal property.

Group table of the quotient group ${\displaystyle \mathrm {D} _{4}/R}$

${\displaystyle \cdot }$	${\displaystyle R}$	${\displaystyle U}$
${\displaystyle R}$	${\displaystyle R}$	${\displaystyle U}$
${\displaystyle U}$	${\displaystyle U}$	${\displaystyle R}$

The elements of the quotient group ${\displaystyle \mathrm {D} _{4}/R}$ are ${\displaystyle R}$ and ${\displaystyle U=f_{\mathrm {v} }R}$. The group operation on the quotient is shown in the table. For example, ${\displaystyle U\cdot U=f_{\mathrm {v} }R\cdot f_{\mathrm {v} }R=(f_{\mathrm {v} }\cdot f_{\mathrm {v} })R=R}$. Both the subgroup ${\displaystyle R=\{\mathrm {id} ,r_{1},r_{2},r_{3}\}}$ and the quotient ${\displaystyle \mathrm {D} _{4}/R}$ are abelian, but ${\displaystyle \mathrm {D} _{4}}$ is not. Sometimes a group can be reconstructed from a subgroup and quotient (plus some additional data), by the semidirect product construction; ${\displaystyle \mathrm {D} _{4}}$ is an example.

The first isomorphism theorem implies that any surjective homomorphism ${\displaystyle \phi \colon G\to H}$ factors canonically as a quotient homomorphism followed by an isomorphism: ${\displaystyle G\to G/\ker \phi \;{\stackrel {\sim }{\to }}\;H}$. Surjective homomorphisms are the epimorphisms in the category of groups.

Every group is isomorphic to a quotient of a free group, in many ways.

For example, the dihedral group ${\displaystyle \mathrm {D} _{4}}$ is generated by the right rotation ${\displaystyle r_{1}}$ and the reflection ${\displaystyle f_{\mathrm {v} }}$ in a vertical line (every element of ${\displaystyle \mathrm {D} _{4}}$ is a finite product of copies of these and their inverses). Hence there is a surjective homomorphism φ from the free group ${\displaystyle \langle r,f\rangle }$ on two generators to ${\displaystyle \mathrm {D} _{4}}$ sending ${\displaystyle r}$ to ${\displaystyle r_{1}}$ and ${\displaystyle f}$ to ${\displaystyle f_{1}}$. Elements in ${\displaystyle \ker \phi }$ are called relations; examples include ${\displaystyle r^{4},f^{2},(r\cdot f)^{2}}$. In fact, it turns out that ${\displaystyle \ker \phi }$ is the smallest normal subgroup of ${\displaystyle \langle r,f\rangle }$ containing these three elements; in other words, all relations are consequences of these three. The quotient of the free group by this normal subgroup is denoted ${\displaystyle \langle r,f\mid r^{4}=f^{2}=(r\cdot f)^{2}=1\rangle }$. This is called a presentation of ${\displaystyle \mathrm {D} _{4}}$ by generators and relations, because the first isomorphism theorem for φ yields an isomorphism ${\displaystyle \langle r,f\mid r^{4}=f^{2}=(r\cdot f)^{2}=1\rangle \to \mathrm {D} _{4}}$.[37]

A presentation of a group can be used to construct the Cayley graph, a graphical depiction of a discrete group.[38]

Many number systems, such as the integers and the rationals, enjoy a naturally given group structure. In some cases, such as with the rationals, both addition and multiplication operations give rise to group structures. Such number systems are predecessors to more general algebraic structures known as rings and fields. Further abstract algebraic concepts such as modules, vector spaces and algebras also form groups.

The group of integers ${\displaystyle \mathbb {Z} }$ under addition, denoted ${\displaystyle \left(\mathbb {Z} ,+\right)}$, has been described above. The integers, with the operation of multiplication instead of addition, ${\displaystyle \left(\mathbb {Z} ,\cdot \right)}$ do not form a group. The associativity and identity axioms are satisfied, but inverses do not exist: for example, ${\displaystyle a=2}$ is an integer, but the only solution to the equation ${\displaystyle a\cdot b=1}$ in this case is ${\displaystyle b={\tfrac {1}{2}}}$, which is a rational number, but not an integer. Hence not every element of ${\displaystyle \mathbb {Z} }$ has a (multiplicative) inverse.[lower-alpha 9]

The desire for the existence of multiplicative inverses suggests considering fractions

$${\displaystyle {\frac {a}{b}}.}$$

Fractions of integers (with ${\displaystyle b}$ nonzero) are known as rational numbers.[lower-alpha 10] The set of all such irreducible fractions is commonly denoted ${\displaystyle \mathbb {Q} }$. There is still a minor obstacle for ${\displaystyle \left(\mathbb {Q} ,\cdot \right)}$, the rationals with multiplication, being a group: because zero does not have a multiplicative inverse (i.e., there is no ${\displaystyle x}$ such that ${\displaystyle x\cdot 0=1}$), ${\displaystyle \left(\mathbb {Q} ,\cdot \right)}$ is still not a group.

However, the set of all nonzero rational numbers ${\displaystyle \mathbb {Q} \smallsetminus \left\{0\right\}=\left\{q\in \mathbb {Q} \mid q\neq 0\right\}}$ does form an abelian group under multiplication, also denoted ${\displaystyle \mathbb {Q} ^{\times }}$.[lower-alpha 11] Associativity and identity element axioms follow from the properties of integers. The closure requirement still holds true after removing zero, because the product of two nonzero rationals is never zero. Finally, the inverse of ${\displaystyle a/b}$ is ${\displaystyle b/a}$, therefore the axiom of the inverse element is satisfied.

The rational numbers (including zero) also form a group under addition. Intertwining addition and multiplication operations yields more complicated structures called rings and – if division by other than zero is possible, such as in ${\displaystyle \mathbb {Q} }$ – fields, which occupy a central position in abstract algebra. Group theoretic arguments therefore underlie parts of the theory of those entities.[lower-alpha 12]

The hours on a clock form a group that uses addition modulo 12. Here, 9 + 4 ≡ 1.

Modular arithmetic for a modulus ${\displaystyle n}$ defines any two elements ${\displaystyle a}$ and ${\displaystyle b}$ that differ by a multiple of ${\displaystyle n}$ to be equivalent, denoted by ${\displaystyle a\equiv b{\pmod {n}}}$. Every integer is equivalent to one of the integers from ${\displaystyle 0}$ to ${\displaystyle n-1}$, and the operations of modular arithmetic modify normal arithmetic by replacing the result of any operation by its equivalent representative. Modular addition, defined in this way for the integers from ${\displaystyle 0}$ to ${\displaystyle n-1}$, forms a group, denoted as ${\displaystyle \mathrm {Z} _{n}}$ or ${\displaystyle (\mathbb {Z} /n\mathbb {Z} ,+)}$, with ${\displaystyle 0}$ as the identity element and ${\displaystyle n-a}$ as the inverse element of ${\displaystyle a}$.

A familiar example is addition of hours on the face of a clock, where 12 rather than 0 is chosen as the representative of the identity. If the hour hand is on ${\displaystyle 9}$ and is advanced ${\displaystyle 4}$ hours, it ends up on ${\displaystyle 1}$, as shown in the illustration. This is expressed by saying that ${\displaystyle 9+4}$ is congruent to ${\displaystyle 1}$ "modulo ${\displaystyle 12}$" or, in symbols,

$${\displaystyle 9+4\equiv 1{\pmod {12}}.}$$

For any prime number ${\displaystyle p}$, there is also the multiplicative group of integers modulo ${\displaystyle p}$.[43] Its elements can be represented by ${\displaystyle 1}$ to ${\displaystyle p-1}$. The group operation, multiplication modulo ${\displaystyle p}$, replaces the usual product by its representative, the remainder of division by ${\displaystyle p}$. For example, for ${\displaystyle p=5}$, the four group elements can be represented by ${\displaystyle 1,2,3,4}$. In this group, ${\displaystyle 4\cdot 4\equiv 1{\bmod {5}}}$, because the usual product ${\displaystyle 16}$ is equivalent to ${\displaystyle 1}$: when divided by ${\displaystyle 5}$ it yields a remainder of ${\displaystyle 1}$. The primality of ${\displaystyle p}$ ensures that the usual product of two representatives is not divisible by ${\displaystyle p}$, and therefore that the modular product is nonzero.[lower-alpha 13] The identity element is represented by ${\displaystyle 1}$, and associativity follows from the corresponding property of the integers. Finally, the inverse element axiom requires that given an integer ${\displaystyle a}$ not divisible by ${\displaystyle p}$, there exists an integer ${\displaystyle b}$ such that

$${\displaystyle a\cdot b\equiv 1{\pmod {p}},}$$

that is, such that ${\displaystyle p}$ evenly divides ${\displaystyle a\cdot b-1}$. The inverse ${\displaystyle b}$ can be found by using Bézout's identity and the fact that the greatest common divisor ${\displaystyle \gcd(a,p)}$ equals ${\displaystyle 1}$.[44] In the case ${\displaystyle p=5}$ above, the inverse of the element represented by ${\displaystyle 4}$ is that represented by ${\displaystyle 4}$, and the inverse of the element represented by ${\displaystyle 3}$ is represented by ${\displaystyle 2}$, as ${\displaystyle 3\cdot 2=6\equiv 1{\bmod {5}}}$. Hence all group axioms are fulfilled. This example is similar to ${\displaystyle \left(\mathbb {Q} \smallsetminus \left\{0\right\},\cdot \right)}$ above: it consists of exactly those elements in the ring ${\displaystyle \mathbb {Z} /p\mathbb {Z} }$ that have a multiplicative inverse.[45] These groups, denoted ${\displaystyle \mathbb {F} _{p}^{\times }}$, are crucial to public-key cryptography.[lower-alpha 14]