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 23 and 22 results in 25 $$ 2^3+2^2=2^{3+2}=2^5 = 10 $$
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 (23) to the second power gives 26 $$ (2^3)^2=2^{3 \cdot2}=2^6 = 12 $$
It's essential to remember that calculating the powers of a group element must adhere to the group's defined operation.
And so forth.
