Group (mathematics)

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A group is a mathematical abstraction consisting of a set of "elements" and an "operation". The operation takes two elements and yield an element. It is perhaps best visualized as addition. The operation must satisfy these properties:

• (closure) The operation is always defined, yielding an element in the set.
• (associativity) Denoting the operation as addition, it satisfies ${\displaystyle x+(y+z)=(x+y)+z}$ for all x, y, and z.
• (identity) There is an "identity element", which we could call "0", such that ${\displaystyle x+0=0+x=x}$ for all x.
• (inverse) Every element has an "inverse element" which, when combined with the element itself, yields the identity. Denoting the inverse of x as -x, we have ${\displaystyle x+(-x)=(-x)+x=0}$ for all x.

The operation is not required to be commutative (${\displaystyle x+y=y+x}$) though, for many groups, it is. Groups for which the operation is commutative are called abelian (uh-BEEL-i-an), in honor of Niels Henrik Abel. But note that "abelian" is not capitalized.

Groups abound throughout mathematics, physics, and related fields. Some are finite and some infinite. Here are some examples:

• The integers, with addition, This is often denoted ${\displaystyle (\mathbb {Z} ,+)\,}$. The identity is zero, and the inverse of x is -x.

• Similarly for the rationals, denoted ${\displaystyle (\mathbb {Q} ,+)}$, the reals, denoted ${\displaystyle (\mathbb {R} ,+)}$, and the complex numbers, denoted ${\displaystyle (\mathbb {C} ,+)}$.

• The nonzero rationals, reals, and complex numbers, with multiplication: ${\displaystyle (\mathbb {Q} ,\cdot )}$, ${\displaystyle (\mathbb {R} ,\cdot )}$, and ${\displaystyle (\mathbb {C} ,\cdot )}$. The identities for these are 1, or the complex number 1+0i. The inverse of an element is its reciprocal, 1/x.

• For any natural number n, the integers modulo n, denoted ${\displaystyle (\mathbb {Z} _{n},+)}$. The elements are the integers {0, 1, ... n-1}, the operation is addition modulo n, and the inverse of x is (-x) mod n. These groups are finite.

• Vector spaces of any dimension, with vector addition.

• For any n, the nonsingular n×n matrices of rational, real, or complex numbers. The operation is matrix multiplication—note that matrix multiplication is not commutative, though it is associative. The identity element is just the identity matrix, and the inverse is the usual matrix inverse. These groups are the "general linear groups", denoted GL(n). They are not abelian if n is greater than 1. The "special linear groups", denoted SL(n), are the groups of matrices with determinant 1.

• For any n, the orthogonal real n×n matrices, or the unitary complex n×n matrices. These groups are important in a number of practical and theoretical areas, and are denoted O(n) and U(n), respectively. The determinant of such a matrix has an absolute value of 1. (For unitary matrices, the determinant is a complex number with modulus 1.) When the determinant is required to be 1, they also form a group, the "special orthogonal" or "special unitary" groups SO(n) and SU(n).

Another very common type of group is a group of "transformations" of some kind, such as the group of rotations, translations, and reflections of some geometrical figure. (This group is called the "Euclidean group".) The group operation for a transformation group is "composition", that is, doing one operation followed by another. The composition operation is usually written with a small circle, as in "${\displaystyle f\circ g\,}$". By convention, this means "do transformation g, then do transformation f", that is, the transformations are done from right to left. The identity of the group is the transformation that does nothing, and the inverse is the transformation that "undoes" a given transformation.

Another famous transformation group is the group of moves on a Rubik's cube.

A very simple and instructive transformation is the permutation (reordering) on N objects. That is, a transformation rearranges the order of some set of N objects. These transformations form the permutation groups, denoted S(n).

For S(2), we have two objects, call them A and B. The possible permutations are:

• Do nothing. This is the identity element of the group, and is of course its own inverse.
• Swap A <=> B. This is the only other group element. It is its own inverse.

For S(3), call the objects A, B, and C. The possible permutations are:

• Do nothing. This is the identity element of the group, and is of course its own inverse.
• Swap A <=> B. This is its own inverse.
• Swap B <=> C. This is its own inverse.
• Swap A <=> C. This is its own inverse.
• Cyclically rearrange: A => B, B => C, and C => A. The inverse of this is the permutation below.
• Cyclically rearrange: C => B, B => A, and A => C. The inverse of this is the permutation above.

As an example of the group operation, if we do A => B, B => C, and C => A, and then swap A <=> B, the result will be that of swapping B and C. If we do the transformations in the other order, the result will be that of swapping A and C. So this group (and all groups S(n) for n greater than 2), is non-abelian.

The groups S(n) have n! (n factorial) elements.

Groups provide a way of giving a precise meaning to symmetry. A symmetry is defined as the group of transformations that leave the geometric figure unchanged. For example, the (near) symmetry of the human face could be characterized by the group consisting of two elements: {no transformation, left-right reflection}. The symmetries of the square are characterized by the group consisting of rotations by a multiple of 90 degrees, and reflections along horizontal, vertical, or diagonal axes. When the details are worked out, this group is seen to have 8 elements.

The group of symmetries of the sphere (that is, all rotations in 3 dimensions) is the group of orthogonal 3x3 matrices with determinant 1. That is, it is SO(3). An interesting theorem relating to eigenvalues says that any such rotation has fixed points, that is, any rotation of the sphere is a rotation around some axis. This theorem is an example of the usefulness of group theory, eigenvectors, and complex numbers to prove interesting geometrical results.