Definition

A group is one of the most fundamental structures in abstract algebra. The concept arises when one isolates the essential properties shared by many mathematical objects: the integers under addition, the nonzero real numbers under multiplication, the symmetries of a geometric figure, and the invertible matrices of a given size all exhibit the same abstract pattern. In formal terms, a group is a set $G$ together with a binary operation $\cdot : G \times G \rightarrow G$ satisfying the following four axioms:

• Closure: for all $a , b \in G$, the element $a \cdot b$ belongs to $G$.
• Associativity: For all $a , b , c \in G$, one has $( a \cdot b ) \cdot c = a \cdot ( b \cdot c )$.
• Identity element: there exists an element $e \in G$ such that $a \cdot e = e \cdot a = a$ for all $a \in G$.
• Inverses: for every $a \in G$ there exists an element $a^{- 1} \in G$ such that $a \cdot a^{- 1} = a^{- 1} \cdot a = e$.

A group $( G , \cdot )$ is called abelian, or commutative, if in addition $a \cdot b = b \cdot a$ for all $a , b \in G$.

The identity element and the inverse of each element are unique. Both facts follow directly from the axioms and are standard early results in any treatment of group theory.

Properties

Several consequences follow from the previous axioms. If $a \cdot b = a \cdot c$ for some $a , b , c \in G$, then multiplying both sides on the left by $a^{- 1}$ gives $b = c$. This is the left cancellation law. The right cancellation law holds by an analogous argument. Together they imply that the equation $a \cdot x = b$ has a unique solution $x = a^{- 1} \cdot b$ in $G$ for any given $a , b \in G$. The inverse of a product satisfies the following identity:

Algebraic hierarchy

Groups are the most elementary objects in the hierarchy of algebraic structures. A group consists of a set equipped with a single binary operation satisfying closure, associativity, the existence of an identity element, and the existence of inverses. When a second operation is introduced and required to distribute over the first, the resulting structure is a ring.

Imposing the further condition that every nonzero element be invertible under multiplication then yields a field. The three structures form a chain of increasing rigidity:

• A group carries one operation with inverses.
• A ring carries two operations, with inverses guaranteed only for addition.
• A field carries two operations, with inverses guaranteed for both addition and all nonzero elements under multiplication.

For example, the integers $\mathbb{Z}$ form a ring but not a field. The rational numbers $\mathbb{Q}$ form a field. Both extend the group structure by adding a second operation.

Order of an element

The order of an element $a$ in a group $G$ is the smallest positive integer $n$ such that $a^{n} = e$, where $e$ is the identity element and the notation $a^{n}$ denotes the product of $a$ with itself $n$ times. If no such integer exists, the element is said to have infinite order. The order of $a$ is denoted $ord ( a )$.

As an example, consider the group $( \mathbb{Z} / 6 \mathbb{Z} , + )$. The element $2$ has order $3$, since $2 + 2 + 2 = 6 \equiv 0 ( mod 6 )$ and neither $2$ nor $2 + 2 = 4$ is congruent to $0$. The element $1$ has order $6$, since one must add $1$ to itself six times to obtain $0$. In the group $( \mathbb{Z} , + )$ every nonzero element has infinite order, because no finite sum of a fixed nonzero integer can equal $0$.

The modulo operator $a mod n$ returns the remainder of the division of $a$ by $n$. For example, $7 mod 5 = 2$ since $7 = 1 \cdot 5 + 2$.

Examples

The set $\mathbb{Z}$ equipped with ordinary addition forms an abelian group. The identity element is $0$, and the inverse of an integer $n$ is $- n$. This is an infinite group and arguably the most natural example of a group in elementary mathematics.

Let $n$ be a positive integer. The set $\mathbb{Z} / n \mathbb{Z} = 0 , 1 , \ldots , n - 1$ equipped with addition modulo $n$ forms a finite abelian group of order $n$. For example, in $\mathbb{Z} / 5 \mathbb{Z}$ one has $3 + 4 = 2$, since $7 \equiv 2 ( mod 5 )$. The identity element is $0$ and the inverse of $k$ is $n - k$.

Let $F$ be a field and let $n$ be a positive integer. The set of all invertible $n \times n$ matrices with entries in $F$, denoted $GL ( n , F )$, forms a group under matrix multiplication. The identity element is the identity matrix $I_{n}$, and the inverse of a matrix $A$ is its matrix inverse $A^{- 1}$. This group is not abelian for $n \geq 2$, since matrix multiplication does not commute in general.

Given a set $1 , 2 , \ldots , n$, a permutation is a bijection from this set to itself. The collection of all such permutations forms a group under composition of functions, denoted $S_{n}$ and called the symmetric group on $n$ elements. The identity element is the identity permutation, and the inverse of a permutation $\sigma$ is the inverse function $\sigma^{- 1}$. The group $S_{n}$ has order $n !$ and is non-abelian for $n \geq 3$.

As a concrete illustration, consider $S_{3}$, which has order $6$. Let $\sigma$ be the permutation sending $1 \rightarrowtail 2$, $2 \rightarrowtail 3$, $3 \rightarrowtail 1$, and let $\tau$ be the permutation sending $1 \rightarrowtail 2$, $2 \rightarrowtail 1$, $3 \rightarrowtail 3$.

To compute $\sigma \circ \tau$, one applies $\tau$ first and then $\sigma$. The element $1$ is sent by $\tau$ to $2$, and then $\sigma$ sends $2$ to $3$, so $1 \rightarrowtail 3$. The element $2$ is sent by $\tau$ to $1$, and then $\sigma$ sends $1$ to $2$, so $2 \rightarrowtail 2$. Finally, $3$ is fixed by $\tau$, and $\sigma$ sends $3$ to $1$, so $3 \rightarrowtail 1$. Thus

An analogous computation yields

Since $\sigma \circ \tau \neq \tau \circ \sigma$, the group $S_{3}$ is indeed non-abelian.

When the axioms fail

A good way to appreciate the group axioms is to look at pairs consisting of a set and an operation that almost form a group, but fail on one specific point. Each failure isolates a different axiom and shows why the definition is cut exactly as it is.

Consider the natural numbers including zero, $\mathbb{N}{0} = 0 , 1 , 2 , \ldots$, equipped with ordinary addition. The operation is closed and associative, and $0$ acts as an identity element. The axiom that fails is the existence of inverses. In fact given any positive integer $n$, there is no element in $\mathbb{N}{0}$ that added to $n$ returns $0$, because the candidate $- n$ lies outside the set. The structure $( \mathbb{N}_{0} , + )$ is therefore not a group, but only a monoid.

The integers with multiplication, $( \mathbb{Z} , \cdot )$, are another example. Closure, associativity, and the identity $1$ are all in place, yet the vast majority of integers lack a multiplicative inverse inside $\mathbb{Z}$. The only elements that admit an inverse are $1$ and $- 1$, since for any other integer $n$ the reciprocal $1 / n$ is not an integer. Dropping all non-invertible elements would leave only the two-element set $1 , - 1$, which is a group under multiplication but a much smaller object than the integers.

The real numbers with multiplication, $( \mathbb{R} , \cdot )$, come even closer to being a group. Every real number different from zero has a multiplicative inverse, namely its reciprocal. The obstacle is a single element: zero has no multiplicative inverse, and its presence in the set is enough to disqualify the whole structure. The fix is to remove it. The set of nonzero reals $\mathbb{R} \backslash 0$ equipped with ordinary multiplication does form an abelian group, with identity $1$ and inverse $a^{- 1} = 1 / a$ for every $a \neq 0$.

These three cases each fail a different axiom. Iinverses for $( \mathbb{N}_{0} , + )$, inverses for all but two elements in $( \mathbb{Z} , \cdot )$, and the existence of an inverse for the single element $0$ in $( \mathbb{R} , \cdot )$. The last case illustrates a recurring pattern in algebra, where removing a problematic element produces a legitimate group.

Cyclic groups

A group $G$ is called cyclic if there exists an element $g \in G$ such that every element of $G$ can be written as a power of $g$, that is:

Such an element $g$ is called a generator of $G$. Every cyclic group is isomorphic either to $\mathbb{Z}$ if it is infinite, or to $\mathbb{Z} / n \mathbb{Z}$ for some positive integer $n$ if it is finite.

The group $( \mathbb{Z} / 6 \mathbb{Z} , + )$ is cyclic with generator $1$, since every element $0 , 1 , 2 , 3 , 4 , 5$ can be obtained as a multiple of $1$. The element $5$ is also a generator, as repeated addition of $5$ modulo $6$ produces all six residues. The element $2$, however, is not a generator, since the multiples of $2$ modulo $6$ are only $0 , 2 , 4$, which form a proper subgroup of $\mathbb{Z} / 6 \mathbb{Z}$.

Subgroups

A subset $H$ of a group $G$ is called a subgroup if $H$ is itself a group under the operation inherited from $G$. Rather than verifying all four group axioms separately, one may use the following criterion: a nonempty subset $H \subseteq G$ is a subgroup of $G$ if and only if for all $a , b \in H$ the element $a \cdot b^{- 1}$ belongs to $H$. This condition encodes closure under the operation and under taking inverses, and the presence of the identity follows from setting $a = b$. One writes $H \leq G$ to indicate that $H$ is a subgroup of $G$.

Every group $G$ has at least two subgroups: the trivial subgroup $e$ and $G$ itself. Any subgroup other than $G$ is called a proper subgroup.

As an example, consider the set of even integers $2 \mathbb{Z} = \ldots , - 4 , - 2 , 0 , 2 , 4 , \ldots$ as a subset of $( \mathbb{Z} , + )$. Taking any two even integers $a = 2 m$ and $b = 2 k$, the inverse of $b$ in $\mathbb{Z}$ is $- b = - 2 k$, so $a + ( - b ) = 2 ( m - k )$, which is again even. The subgroup criterion is therefore satisfied, and $2 \mathbb{Z}$ is a subgroup of $\mathbb{Z}$.

Group homomorphisms and isomorphisms

A group homomorphism is a function between two groups that preserves the group structure. Given two groups $( G , \cdot )$ and $( H , \star )$, a function $\varphi : G \rightarrow H$ is a homomorphism if for all $a , b \in G$ holds:

This condition requires that applying $\varphi$ after performing the operation in $G$ yields the same result as first applying $\varphi$ to each element and then performing the operation in $H$. Several basic properties follow from this definition. A homomorphism $\varphi : G \rightarrow H$ necessarily maps the identity of $G$ to the identity of $H$, and satisfies $\varphi ( a^{- 1} ) = \varphi ( a )^{- 1}$ for all $a \in G .$ Two particularly important subsets associated with a homomorphism $\varphi : G \rightarrow H$ are the kernel and the image. The kernel is defined as:

In this case $e_{H}$ denotes the identity of $H$, and the image is defined as:

The kernel is always a subgroup of $G$, and the image is always a subgroup of $H$. Moreover, a homomorphism is injective if and only if its kernel contains only the identity element of $G .$

A homomorphism $\varphi : G \rightarrow H$ that is injective and surjective is called an isomorphism. When an isomorphism exists between $G$ and $H$, the two groups are said to be isomorphic, written $G \cong H$. Isomorphic groups are structurally identical. They differ only in the names of their elements and their operation, not in any property that is intrinsic to their group structure. As an example, consider the group $( \mathbb{Z} / 2 \mathbb{Z} , + )$ and the group $( 1 , - 1 , \times )$ under ordinary multiplication. Define $\varphi : \mathbb{Z} / 2 \mathbb{Z} \rightarrow 1 , - 1$ by setting $\varphi ( 0 ) = 1$ and $\varphi ( 1 ) = - 1$. Since:

the function $\varphi$ preserves the group operation. As it is also bijective, it is an isomorphism, and therefore $\mathbb{Z} / 2 \mathbb{Z} \cong { 1 , - 1 }$.

isomorphismimagekernelhomomorphismsubgroup criterionsubgroupgroup notationinfinite groupsfinite groupsgeneratorcyclic grouporder of elementorder of groupabelian groupinverse elementidentity elementassociativityclosurebinary operationgroupmorphismsstructuredefinition