- This article is about the basic notions. For advanced topics, see Group theory.
In mathematics, a group is a kind of algebraic structure. A group has a set and an operation. The group's operation can put together any two elements of the group's set to make a third element, also in the set. A familiar example of a group is the set of integers with the addition operator.
Instead of "an element of the group's set", mathematicians usually save words by saying "an element of the group".
Mathematicians use capital letters to stand for groups. They often use G, H, or K.
They also use lower-case letters to stand for group elements. For example, they would say "a is in G" to mean "a is an element of G".
They write group operations with symbols like • or *, or by writing two elements next to each other. So "a • b", "a * b", and "ab" can all mean "the element formed when the group's operation combines a and b".
Not every set and operation make a group. A group's set and operation must obey some special rules. These are called group axioms. The following list has each axiom in both words and mathematical symbols.
- Closure: When a group's operation puts together two elements, the element that is made must also be part of the group.
- For all a, b in G, a • b is also in G.
- Identity element: One element of the group is special. It is called the identity element. If the operation puts together the identity element and any second element, the answer will be that second element. The identity element is usually written as e. Sometimes the identity element is written as 0 or 1.
- There exists e in G, so that for all a in G,
e • a = a • e = a.
- Associativity: When the operation is used twice to combine three elements, it does not matter in what order they are done, because the answer will be the same.
- For all a, b and c in G, (a • b) • c = a • (b • c).
- Inverse element: Every element in the group has another element in the group that is called its inverse. When the operation combines any element and its inverse, the answer is the identity element. Sometimes, the inverse is indicated by writing either a '-' in front, or by writing -1 in superscript, so the inverse of the element a is either -a or a-1.
- For each a in G, there must exist b in G such that a • b = b • a = e, where e is G's identity element.
Some more ideasEdit
To name the group, one should say all three parts: (G, •, e). But when it is clear, one may just say that G is a group.
It is possible to find that for two elements a and b in a group, a • b is different from b • a. For example, maybe a • b = c, but b • a = d.
The number of elements in a group is called the group's order.
Examples of groupsEdit
Integers, addition, zeroEdit
One everyday example of a group is the set of integers with the addition operation and zero (the identity). It is easy to see that when adding two integers, the result is always an integer. So closure is true. The identity element for this group is zero. Because the order of additions does not matter (in other words, ), associativity is true. And the inverse of any integer is its opposite number. This group is also an abelian group. It is an infinite group (also called a group of infinite order) because it has an infinite number of elements: the integers.
The set of integers with the multiplication operation is not a group, because with multiplication, most integers have no inverse that is also an integer.
Another important example of a group is a group of symmetries, where each symmetry of an object is an element of the group. One can also think of this group as the set of bijections on n elements—with function composition as the operation. The symmetric group on n elements is often written as .
The operation of the group is this: make the change of the one symmetry and then make the change of the other symmetry. The combination is also a symmetry, and that means that there is closure.
The identity element is the symmetry in which nothing changes.
Any symmetric change can be undone by reversing the change. That reverse is the inverse element.
Symmetry groups provide many examples of groups. For example, here are all of the symmetries of a square: turn the square clockwise 90o, 180o, or 270o; turn the square counter-clockwise 90o, 180o, or 270o; reflect the image of the square in a vertical mirror, or in a horizontal mirror, or a diagonal mirror to the right or the left; or leave the square unchanged. This is a finite group (or a group of finite order). It is a non-abelian group because it can make a difference which order when making a turn and a reflection.
It can be proved that every group of finite order is a group of symmetries. For this reason, group theory is called the mathematical study of symmetry.
Many things in mathematics are groups. For example, equations. The ways of solving equations can be a group. Physical things can be studied as groups. There are important symmetries about physical things which are groups.
Sometimes, part of the group is also a group. In that case, the part is called a subgroup. For example, there is this subgroup of the group of symmetries of a square: only the turns of the square and the identity (not the reflections in a mirror). Many useful things can be proved about groups and their subgroups.
- "Comprehensive List of Algebra Symbols". Math Vault. 2020-03-25. Retrieved 2020-10-12.
- "group theory | Definition, Axioms, & Applications". Encyclopedia Britannica. Retrieved 2020-10-12.
- Weisstein, Eric W. "Group". mathworld.wolfram.com. Retrieved 2020-10-12.