Exponentiation

Exponentiation is a mathematical operation, written as bⁿ, involving two numbers, the base b and the exponent n. When n is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, bⁿ is the product of multiplying n bases:

In that case, bⁿ is called the n-th power of b, or b raised to the power n.

The exponent is usually shown as a superscript to the right of the base. Some common exponents have their own names: the exponent 2 (or 2nd power) is called the square of b (b²) or b squared; the exponent 3 (or 3rd power) is called the cube of b (b³) or b cubed. The exponent −1 of b, or 1 / b, is called the reciprocal of b.

When n is a negative integer and b is not zero, bⁿ is naturally defined as 1/b⁻ⁿ, preserving the property bⁿ × b^m = b^{n + m}.

The definition of exponentiation can be extended to allow any real or complex exponent. Exponentiation by integer exponents can also be defined for a wide variety of algebraic structures, including matrices.

Exponent (disambiguation)

Exponent may refer to:

See also

Torsion group

In group theory, a branch of mathematics, a torsion group or a periodic group is a group in which each element has finite order. All finite groups are periodic. The concept of a periodic group should not be confused with that of a cyclic group.

The exponent of a periodic group G is the least common multiple, if it exists, of the orders of the elements of G. Any finite group has an exponent: it is a divisor of |G|.

Burnside's problem is a classical question, which deals with the relationship between periodic groups and finite groups, if we assume only that G is a finitely-generated group. The question is whether specifying an exponent forces finiteness (to which the answer is 'no', in general).

Examples of infinite periodic groups include the additive group of the ring of polynomials over a finite field, and the quotient group of the rationals by the integers, as well as their direct summands, the Prüfer groups. Another example is the union of all dihedral groups. None of these examples has a finite generating set, and any periodic linear group with a finite generating set is finite. Explicit examples of finitely generated infinite periodic groups were constructed by Golod, based on joint work with Shafarevich, see Golod–Shafarevich theorem, and by Aleshin and Grigorchuk using automata.