Power of an Element in a Group

In any group $ (G, *) $, when you take an element $g$, its $n$-th power is determined by applying the group's * operation on $g$ repeatedly, $n$ times. Essentially, $$ g^n = \underbrace{ g*g*g \ *...* \ g }_{n \ times} $$ showcases the operation.

The operation * itself is dependent on the nature of the group at hand. This could be addition for additive groups, multiplication for multiplicative ones, and so forth.

More formally, for any positive integer $n$, the definition of $g^n$ goes as follows:

$$ g^1 = g $$

$$ g^n = g \ \cdot \ ... \ \cdot g \ \ \ \ for \ all \ n > 1 $$

Here, "$\cdot$" represents the group's operation in action.

In additive groups, for example, we might replace $\cdot$ with $+$ and refer to $ng$ as summing up $g$ with itself $n$ times.

$$ g^1 = g $$

$$ g^n = g \ + \ ... \ + \ g \ \ \ \ for \ \ all \ n > 1 $$

When dealing with negative integers $n$, and if $g$ has an inverse $g^{-1}$, then $g^n$ (for $n < 0$) is understood as $(g^{-1})^{|n|}$, implying the use of $g$'s inverse, multiplied by itself $|n|$ times.

$$ g^{n} = g^{-1} \ \cdot \ ... \ \cdot g^{-1} \ \ \ \ for \ all \ n < 0 $$

At $n = 0$, $g^0$ is simply the group's identity element $ e $, which, when combined with any group element through the group operation, yields that very element back.

$$ g^0 = e $$

In the realm of multiplicative groups, the identity element is commonly 1, while in additive groups, it's 0.

Power Operations

Power operations adhere to the same principles as in conventional algebra.

$$ g^n*g^m = g^{m+n} = \underbrace{g*g* \ ... \ *g}_{m+n} $$

$$ (g^n)^m = g^{m \cdot n} = \underbrace{g*g* \ ... \ *g}_{m \cdot n} $$

Naturally, the operation to be repeated is the group's own operation.

Understanding the power of a group element is crucial for analyzing cyclic structures within groups and grasping how certain elements can create subgroups within larger ones. A group formed by a single element $g$ through its powers is known as a cyclic group, with $g$ serving as the group's generator.

Illustrative Example

Take the group $(\mathbb{Z}, +)$, where $\mathbb{Z}$ stands for all integers and $+$ the operation of addition.

This exemplifies an additive group, meaning "powers" in this context actually refer to the act of repeatedly summing an element.

The identity element here is $0$, as adding any integer to $0$ returns the original integer.

Positive Power Example:

Consider the element $3$ in $\mathbb{Z}$.

The 4th power of $3$, though written as $3^4$ in multiplicative terms, translates to the additive sequence of $3$ added to itself 4 times. Hence, we arrive at:

$$ 3 + 3 + 3 + 3 = 12 $$

Thought of additively, this is akin to $4 \times 3 = 12$.

Negative Exponent Example:

Moving to the element $ 4 $ with an intent to compute $ 4^{-3} $ in the group $ (Z,+) $.

With -4 being the additive inverse of 4, we find:

$$ (-4)^{-3} = (-4) + (-4) + (-4) = -12 $$

Thus, in this instance, $ 4^{-3} $ in the group $ (Z,+) $ effectively means adding -4 to itself three times.

Case for $n = 0$:

For any element within $\mathbb{Z}$, the "0-th power" aligns with the group's identity element, $0$.

Accordingly, no matter the chosen element, $g^0 = 0$ in this additive framework.

As an example, $ 4^0 $ in the additive group $ (Z,+) $ equals zero.

$$ 4^0 = 0 $$

Here, 0 serves as the identity element of the additive group.

Conversely, $ 4^0 $ in a multiplicative group $ (Z, \cdot) $ would be one.

$$ 4^0 = 1 $$

Here, 1 is recognized as the identity element of the multiplicative group.

Examples of Power Operations:

In the additive group $ ( Z, + ) $, the expression $ 2^3 × 2^2 $ translates to the power $ 2^{3+2} = 2^5 $

$$ 2^3 × 2^2 = 2^5 = 2 + 2 + 2 + 2 + 2 = 10 $$

Conversely, the expression $ ( 2^3 )^2 $ equates to the power $ 2^{3 \cdot 2} = 2^6 $

$$ (2^3)^2 = 2^6 = 2 + 2 + 2 + 2 + 2 + 2 = 12 $$

Here too, the operation being repeated is specific to the group, with addition being the operation in an additive group.

Generating a Cyclic Group from an Element:

Observing the element $1$ in $\mathbb{Z}$, we see that through continuous addition of this element, the entire group $\mathbb{Z}$ can be produced. For instance:

$1^1 = 1$
$1^2 = 1 + 1 = 2$
$1^3 = 1 + 1 + 1 = 3$

Proceeding in this manner, and by employing the inverse of $1$, which is $-1$, all negative integers can be generated.

Hence, $1$ (and $-1$) acts as a generator for the group $\mathbb{Z}$ under addition, rendering $\mathbb{Z$ an exemplar of an infinite cyclic group.