Index Laws

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Theorem

Index Laws for Semigroup

Sum of Indices

Let $\struct {S, \circ}$ be a semigroup.

For $a \in S$, let $\circ^n a = a^n$ be defined as the $n$th power of $a$:

$a^n = \begin{cases} a & : n = 1 \\ a^x \circ a & : n = x + 1 \end{cases}$

That is:

$a^n = \underbrace {a \circ a \circ \cdots \circ a}_{n \text{ copies of } a} = \circ^n \paren a$

Then:

$\forall m, n \in \N_{>0}: a^{n + m} = a^n \circ a^m$


Product of Indices

Let $\struct {S, \circ}$ be a semigroup.

For $a \in S$, let $\circ^n a = a^n$ be the $n$th power of $a$.


Then:

$\forall m, n \in \N_{>0}: a^{n m} = \paren {a^n}^m = \paren {a^m}^n$


Index Laws for Monoid

Sum of Indices

Let $\struct {S, \circ}$ be a monoid whose identity element is $e$.

For $a \in S$, let $\circ^n a = a^n$ be defined as the $n$th power of $a$:

$a^n = \begin{cases} e & : n = 0 \\ a^x \circ a & : n = x + 1 \end{cases}$

That is:

$a^n = \underbrace {a \circ a \circ \cdots \circ a}_{n \text{ copies of } a} = \map {\circ^n} a$

while:

$a^0 = e$


Then:

$\forall m, n \in \N: a^{n + m} = a^n \circ a^m$


Product of Indices

Let $\struct {S, \circ}$ be a monoid whose identity element is $e$.

For $a \in S$, let $\circ^n a = a^n$ be the $n$th power of $a$.


Then:

$\forall m, n \in \N: a^{n m} = \paren {a^n}^m = \paren {a^m}^n$


Powers of Group Elements

Let $\struct {G, \circ}$ be a group whose identity is $e$.

Let $g \in G$.


Then the following results hold:

Negative Index

$\forall n \in \Z: \paren {g^n}^{-1} = g^{-n} = \paren {g^{-1} }^n$


Sum of Indices

$\forall m, n \in \Z: g^m \circ g^n = g^{m + n}$


Product of Indices

$\forall m, n \in \Z: \paren {g^m}^n = g^{m n} = \paren {g^n}^m$


Index Laws for Field

Let $\struct {F, +, \circ}$ be a field with zero $0_F$ and unity $1_F$.

Let $F^* = F \setminus {0_F}$ denote the set of elements of $F$ without the zero $0_F$.


Then the following hold:

Common Index

$(a):\quad \forall a, b \in \F^* : \forall n \in \Z : a^n \circ b^n = \paren{ab}^n$
$(b):\quad \forall a, b \in \F : \forall n \in \Z_{\ge 0} : a^n \circ b^n = \paren{ab}^n$

Negative Index

$\forall a \in \F^* : \forall n \in \Z : \paren{a^n}^{-1} = a^{-n} = \paren{a^{-1}}^n$

Sum of Indices

$(a):\quad \forall a \in \F^* : \forall n, m \in \Z : a^m \circ a^n = a^\paren{m + n}$
$(b):\quad \forall a \in \F : \forall n, m \in \Z_{\ge 0} : a^m \circ a^n = a^\paren{m + n}$

Product of Indices

$(a):\quad \forall a \in \F^* : \forall n, m \in \Z : \paren{a^m}^n = a^\paren{mn}$
$(b):\quad \forall a \in \F : \forall n, m \in \Z_{\ge 0} : \paren{a^m}^n = a^\paren{mn}$

Source of Name

The name index laws originates from the name index to describe the exponent $y$ in the power $x^y$.