Complex Numbers Exponential - Generalizations - in Abstract Algebra

In Abstract Algebra

Exponentiation for integer exponents can be defined for quite general structures in abstract algebra.

Let X be a set with a power-associative binary operation which is written multiplicatively. Then xn is defined for any element x of X and any nonzero natural number n as the product of n copies of x, which is recursively defined by

\begin{align} x^1 &= x \\ x^n &= x^{n-1}x \quad\hbox{for }n>1 \end{align}

One has the following properties

\begin{align} (x^i x^j) x^k &= x^i (x^j x^k) \quad\text{(power-associative property)} \\ x^{m+n} &= x^m x^n \\ (x^m)^n &= x^{mn} \end{align}

If the operation has a two-sided identity element 1 (often denoted by e), then x0 is defined to be equal to 1 for any x.

\begin{align} x1 &= 1x = x \quad\text{(two-sided identity)} \\ x^0 &= 1 \end{align}

If the operation also has two-sided inverses, and multiplication is associative then the magma is a group. The inverse of x can be denoted by x−1 and follows all the usual rules for exponents.

\begin{align} x x^{-1} &= x^{-1} x = 1 \quad\text{(two-sided inverse)} \\ (x y) z &= x (y z) \quad\text{(associative)} \\ x^{-n} &= \left(x^{-1}\right)^n \\ x^{m-n} &= x^m x^{-n} \end{align}

If the multiplication operation is commutative (as for instance in abelian groups), then the following holds:

If the binary operation is written additively, as it often is for abelian groups, then "exponentiation is repeated multiplication" can be reinterpreted as "multiplication is repeated addition". Thus, each of the laws of exponentiation above has an analogue among laws of multiplication.

When one has several operations around, any of which might be repeated using exponentiation, it is common to indicate which operation is being repeated by placing its symbol in the superscript. Thus, x∗n is x ∗ ··· ∗ x, while x#n is x # ··· # x, whatever the operations ∗ and # might be.

Superscript notation is also used, especially in group theory, to indicate conjugation. That is, gh = h−1gh, where g and h are elements of some group. Although conjugation obeys some of the same laws as exponentiation, it is not an example of repeated multiplication in any sense. A quandle is an algebraic structure in which these laws of conjugation play a central role.

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