Group (mathematics)

In mathematics, a group is a special collection of items, called a set, with a way to combine any two items to get another item in the same collection. This combining method must follow some important rules: it must work in the same way no matter how you group the items, there must be a special item that doesn’t change anything when combined with others, and every item must have a matching item that can undo the combination. For example, the integers with the addition operation form a group.

The idea of a group helps us understand many different mathematical structures, like numbers, geometric shapes, and solutions to equations. Because groups are so useful in many areas of math and science, some people think they are one of the most important ideas in modern mathematics.

In geometry, groups help us study symmetries and changes to shapes. The ways a shape can be moved and still look the same form a group called the symmetry group of the shape. Groups also appear in physics, where they describe the symmetries of space and time, and in chemistry, where they describe how molecules are arranged. The idea of groups started when mathematicians were solving equations with many solutions, and it has grown into a big and active area of study.

Definition and illustration

First example: the integers

The integers, which include numbers like -4, -3, -2, -1, 0, 1, 2, 3, 4, and so on, form a group when we use addition. For any two integers, their sum is also an integer. This means the operation of addition combines any two integers to produce another integer.

Addition has special properties that make the integers a group:

Associativity: When we add three numbers, it doesn’t matter how we group them. For example, (a + b) + c is the same as a + (b + c).
Identity element: The number 0 is the identity element because adding 0 to any integer leaves the integer unchanged. For example, a + 0 = a.
Inverse element: Every integer has an inverse, which is the number that, when added to the original, gives 0. For example, the inverse of a is -a because a + (-a) = 0.

Definition

A group is a set of elements together with an operation that combines any two elements to form a new element. This operation must satisfy three important rules, called the group axioms:

Associativity: The way we group the elements when combining three or more doesn’t change the result.
Identity element: There is a special element that, when combined with any other element, leaves that element unchanged.
Inverse element: Every element has a partner that, when combined with it, gives the identity element.

Second example: a symmetry group

A square has eight symmetries, which are ways to move the square that leave it looking the same. These symmetries include leaving the square unchanged, rotating it by 90°, 180°, or 270°, and reflecting it across different lines. These symmetries form a group under the operation of combining two symmetries to make a new one.

For example, rotating a square by 90° and then reflecting it horizontally gives a different result than reflecting it horizontally and then rotating by 90°. This shows that the order of operations can matter. Each symmetry has an inverse, meaning doing the symmetry and then its inverse brings the square back to its original position.

History

Main article: History of group theory

The idea of a group started when mathematicians tried to solve hard equations. A French mathematician named Évariste Galois showed how the way solutions could be moved around each other could tell if an equation could be solved. Later, mathematicians used groups to study shapes and numbers.

Groups became a big part of math when mathematicians like Sophus Lie and Ferdinand Georg Frobenius began studying them more deeply. By the 1960s, mathematicians worked together to classify all the simplest kinds of groups, a huge project that took many years to complete. Today, group theory helps in many areas of math and science.

Elementary consequences of the group axioms

Basic facts about all groups can be learned directly from the rules that define them. For example, applying a rule called "associativity" many times shows that when we combine more than two elements, the order in which we put parentheses (like parentheses in arithmetic) does not matter. This means we can usually leave out the parentheses.

The rules also tell us that the identity element is unique. This means there is only one identity element in a group. Any two identity elements must be the same.

The rules also show that each element has a unique inverse. This means for any element, there is only one element that can "undo" it when combined with it.

For any two elements in a group, there is a unique way to solve the equation a ⋅ x = b for x, and it is given by a⁻¹ ⋅ b. Similarly, the unique solution to x ⋅ a = b is b ⋅ a⁻¹. These facts help us understand how elements interact within the group.

Basic concepts

Groups are a special kind of set in mathematics. A set is a collection of objects, and a group has a rule for combining any two objects in the set to make a new one, following certain rules.

For a group to work, the combination rule needs to meet three main conditions:

Associativity: When you combine three objects, the order in which you combine them doesn’t change the result.
Identity element: There is a special object that, when combined with any other object, leaves it unchanged.
Inverse elements: For every object, there is another object that, when combined with it, gives the identity element.

An example of a group is the set of whole numbers (like 0, 1, 2, 3...) with the operation of addition. Adding any two whole numbers gives another whole number, and the rules above are all satisfied.

Cayley table of the quotient group D 4 / R {\displaystyle \mathrm {D} _{4}/R}
⋅ {\displaystyle \cdot }	R {\displaystyle R}	U {\displaystyle U}
R {\displaystyle R}	R {\displaystyle R}	U {\displaystyle U}
U {\displaystyle U}	U {\displaystyle U}	R {\displaystyle R}

Examples and applications

Groups are found in many areas of mathematics. For example, the set of whole numbers with addition forms a group. When considering multiplication instead of addition, we get different types of groups called multiplicative groups. These ideas help us understand more complex mathematical structures.

Groups are also useful in studying shapes and spaces. For instance, Henri Poincaré used groups to study shapes, which helped connect properties of shapes to group properties. In chemistry, groups help describe the symmetry of molecules, and in physics, they help predict how materials change under certain conditions.

Numbers

Many number systems, like whole numbers and fractions, naturally form groups. For example, whole numbers with addition form a group. Fractions with multiplication also form a group, as long as we exclude zero. These number systems are building blocks for more advanced mathematical concepts.

Modular arithmetic

Modular arithmetic deals with numbers in cycles, like the hours on a clock. Adding numbers in a cycle, such as adding hours on a clock, forms a group. For prime numbers, we can also create groups using multiplication in a similar cyclic way.

Cyclic groups

A cyclic group is one where every element can be generated by repeating a single element. For example, in the group of numbers on a clock, adding one hour repeatedly generates all the hours. These groups are common and help simplify many mathematical problems.

Symmetry groups

Symmetry groups study the symmetries of objects, like the symmetries of a square or the patterns in crystals. These groups help us understand the properties of shapes and materials by looking at how they can be transformed while still looking the same.

General linear group and representation theory

Matrix groups are groups made from matrices, which are grids of numbers. These groups are important in computer graphics and many areas of mathematics. Representation theory helps us understand groups by showing how they act on spaces, making abstract concepts more concrete.

Galois groups

Galois groups help solve equations by studying their symmetries. For example, the solutions to quadratic equations can be understood using group theory. These groups also help us understand when equations can be solved using basic operations and roots.


Buckminsterfullerene displays icosahedral symmetry	Ammonia, NH₃. Its symmetry group is of order 6, generated by a 120° rotation and a reflection.	Cubane C₈H₈ features octahedral symmetry.	The tetrachloroplatinate(II) ion, [PtCl₄]²⁻ exhibits square-planar geometry

Finite groups

Main article: Finite group

A finite group is a group that has a limited number of elements. This number is called the order of the group. One important type of finite group is the symmetric group, which consists of all possible ways to rearrange, or permute, a set of objects.

For example, if you have three letters—A, B, and C—you can arrange them in six different ways: ABC, ACB, BAC, BCA, CAB, and CBA. This collection of arrangements forms a symmetric group. The "operation" in this group is combining two arrangements to get a new one, and there's always one arrangement that leaves everything in its original place (the identity element).

Every finite group is connected to symmetric groups in a special way, showing how these arrangements help us understand all finite groups.

Groups with additional structure

A group is a set of items with a special way to combine any two items, called an operation. This operation must follow certain rules: it must work in any order (associative), there must be a "do nothing" item (identity element), and every item must have a matching "undo" item (inverse element).

This idea can be expanded to include more structure. For example, some groups also have a shape or space structure, called topological groups. These groups work well with ideas from geometry and can be used to study patterns in numbers and shapes. Another example is Lie groups, which combine group rules with ideas from smooth, curved spaces. These are very important in physics, helping to describe how things move and change in space and time.

Generalizations

More general structures can be made by relaxing some of the rules that define a group. For instance, if we stop requiring that every element has an opposite (inverse), the resulting structure is called a monoid. The natural numbers (including zero) with addition form a monoid, as do the nonzero integers with multiplication.

A group can also be viewed as a special kind of small category with one object where every map between the object is reversible. More broadly, a groupoid is a small category where every map is reversible, though not all such categories form a group.

Finally, these ideas can be extended by using operations that take more than two inputs instead of just two, leading to the concept of an n-ary group.

Examples
Set	Natural numbers ⁠ N {\displaystyle \mathbb {N} } ⁠		Integers ⁠ Z {\displaystyle \mathbb {Z} } ⁠		Rational numbers ⁠ Q {\displaystyle \mathbb {Q} } ⁠ Real numbers ⁠ R {\displaystyle \mathbb {R} } ⁠ Complex numbers ⁠ C {\displaystyle \mathbb {C} } ⁠				Integers modulo 3 ⁠ Z / n Z = { 0 , 1 , 2 } {\displaystyle \mathbb {Z} /n\mathbb {Z} =\{0,1,2\}} ⁠
Operation	+	×	+	×	+	−	×	÷	+	×
Total	yes	yes	yes	yes	yes	yes	yes	no	yes	yes
Identity	yes	yes	yes	yes	yes	no	yes	no	yes	yes
Inverse	no	no	yes	no	yes	no	only if ⁠ a ≠ 0 {\displaystyle a\neq 0} ⁠	no	yes	only if ⁠ a ≠ 0 {\displaystyle a\neq 0} ⁠
Divisibility	no	no	yes	no	yes	yes	only if ⁠ a ≠ 0 {\displaystyle a\neq 0} ⁠	only if ⁠ a ≠ 0 {\displaystyle a\neq 0} ⁠	yes	no
Associative	yes	yes	yes	yes	yes	no	yes	no	yes	yes
Commutative	yes	yes	yes	yes	yes	no	yes	no	yes	yes
Structure	monoid	monoid	abelian group	monoid	abelian group	quasigroup	monoid	quasigroup (with 0 removed)	abelian group	monoid