# Power of a Group Element

In the context of group theory, the concept of an **element's power**, $ g \in G $, within a group (G,*) involves applying the group's operation, *, to that element repeatedly for n times. $$ g^n = \underbrace{g*g*...*g}_{n \ times} $$

This highlights that the operation defined by the group is pivotal when discussing an element's power.

The effect of this operation can vary greatly depending on the group's nature.

**Additive Groups**

For an additive group (G,+), where the operation is addition, powering an element, say g to the third, means adding g to itself three times. $$ g^3 = g+g+g = 3g $$ For example, within the integer group (Z,+) using addition: $$ 2^3=2+2+2=6 $$**Multiplicative Groups**

In a multiplicative group (G,·), where multiplication is the operation, the third power of an element, g, involves multiplying g by itself three times. $$ g^3 = g \cdot g \cdot g $$ For instance, in the integer group (Z,*) with multiplication: $$ 2^3=2 \cdot 2 \cdot 2 = 8 $$

Therefore, the outcome is inherently linked to the group's operation.

**Note**: This explanation uses additive and multiplicative groups for simplicity, but the principles apply universally across different groups and operations beyond just addition and multiplication.

When the **power's exponent is zero**, the outcome is invariably the group's identity element.

In an additive group (G,+), for instance, any element raised to the power of zero equals the group's identity element, which is zero in this case.

$$ g^0 = 0 $$

**Example**: For the integer group (Z,+) with addition, two raised to the power of zero is zero $$ 2^0=0 $$ This principle applies to any element within the group.

Conversely, in a multiplicative group (Z,·), any element raised to zero equals 1, since 1 is the identity element for multiplication.

$$ g^0 = 1 $$

**Example**: In the integer group (Z,·) under multiplication, two raised to zero equals 1 because 1 is the identity element for multiplication: $$ 2^0=1 $$ This rule holds for any element raised to zero.

### Calculating Powers with Negative Exponents

If the exponent is negative, one must apply the group operation (*) n times to the inverse element.

$$ g^{n} = \underbrace{g^{-1}*g^{-1}*...*g^{-1}}_{n \ times} \ \ \ with \ \ n<0 $$

For instance, in the additive group (Z,+), the power 2^{-3} involves adding the additive inverse of 2, which is -2, three times. $$ 2^{-3} = (-2)+(-2)+(-2)=-6 $$ Conversely, in the multiplicative group (Q,·) of rational numbers, the power 2^{-3} results from thrice multiplying the multiplicative inverse of 2, which is 1/2. $$ 2^{-3} = \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{8} $$

### Power Operations

The algebraic operations applicable to powers within group theory are consistent with general power rules.

Multiplying two powers that share the same base equates to a single power of that base with an exponent that is the sum of the two exponents.

$$ a^n * a^m = a^{n+m} $$

For example, in the additive group (Z,+), combining powers like 2^{3} and 2^{2 } results in 2^{5} $$ 2^3+2^2=2^{3+2}=2^5 = 10 $$

Raising a power by another exponent results in multiplying those exponents together.

$$ (a^n)^m = a^{n \cdot m} $$

For instance, in the additive group (Z,+), raising (2^{3}) to the second power gives 2^{6} $$ (2^3)^2=2^{3 \cdot2}=2^6 = 12 $$

It's essential to remember that calculating the powers of a group element must adhere to the group's defined operation.

And so forth.