# Category:Powers (Abstract Algebra)

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This category contains results about **Powers (Abstract Algebra)**.

Definitions specific to this category can be found in Definitions/Powers (Abstract Algebra).

Let $\struct {S, \circ}$ be a magma which has no identity element.

Let $a \in S$.

Let the mapping $\circ^n a: \N_{>0} \to S$ be recursively defined as:

- $\forall n \in \N_{>0}: \circ^n a = \begin{cases}

a & : n = 1 \\ \paren {\circ^r a} \circ a & : n = r + 1 \end{cases}$

The mapping $\circ^n a$ is known as the **$n$th power of $a$ (under $\circ$)**.

## Pages in category "Powers (Abstract Algebra)"

The following 20 pages are in this category, out of 20 total.

### P

- Power of Element of Semigroup
- Power of Element/Semigroup
- Power of Identity is Identity
- Power of Product of Commuting Elements in Monoid equals Product of Powers
- Power of Product of Commuting Elements in Semigroup equals Product of Powers
- Powers of Commuting Elements of Monoid Commute
- Powers of Commuting Elements of Semigroup Commute
- Powers of Elements in Group Direct Product
- Powers of Field Elements Commute
- Powers of Group Element Commute
- Powers of Group Elements
- Powers of Group Elements/Product of Indices
- Powers of Group Elements/Sum of Indices
- Powers of Semigroup Element Commute
- Product of Indices Law for Field
- Product of Powers of Group Elements