Cyclic Subgroups
A cyclic subgroup within a group (G,*) is defined as a subgroup generated by a single element g∈G. $$ \langle g \rangle = \{g^n | n \in \mathbb{Z}\} $$ The element g is known as the generator of the cyclic subgroup.
This means that with a given group operation (*) and an element g in G, the cyclic subgroup <g> generated by g comprises all elements expressible as gn for any integer n.
The notation commonly used to represent the cyclic subgroup generated by g is <g>.
$$ \langle g \rangle = \{g^n | n \in \mathbb{Z}\} $$
In this expression, gn signifies the element g multiplied by itself n times according to the group's * operation, with n being any integer.
Note: For n<0, meaning n is negative, one must multiply the inverse of gn |n| times by the group operation. If n=0, then g0=e corresponds to the neutral (identity) element (e) of the group (G,*). For an in-depth look at computing the power of an element within a group, follow the link.
Cyclic subgroups can be categorized as either finite or infinite, depending on their size.
- Finite Subgroup
A subgroup is finite if there is a smallest positive integer n such that gn equals the group's neutral (identity) element, gn=e. In this instance, n is referred to as the order of g and consequently the order of the cyclic subgroup <g>. - Infinite Subgroup
A subgroup is infinite if no smallest positive integer n results in gn equaling the neutral (identity) element of the group. Thus, the order of g and by extension, the subgroup <g>, is considered infinite.
An Illustrative Example
Consider the additive group (Z,+), where Z denotes the set of all integers, and the group operation is addition (+).
$$ (G,+) $$
For this example, let's select g=3 as the generator of our subgroup.
The cyclic subgroup generated by 3 is denoted <3> and includes all integer multiples of 3.
$$ \langle 3 \rangle = \{ \ldots, -9, -6, -3, 0, 3, 6, 9, 12, \ldots \} $$
In this scenario, to "multiply" the number g=3 by an integer n effectively means to add 3 to itself n times.
The number 3 serves as the cyclic subgroup <3>'s generator within the integer set Z.
Hence, each member of the subgroup <3> can be derived by either adding or subtracting 3 from itself a specific number of times.
Example: The operation 34, in the context of addition (+), translates to multiplying 3 by four times.
$$ 3^4 = 3+3+3+3 = 12 $$
It's important to understand that gn is not a numeric power but rather a notation for performing the group operation, which in this case is addition (+), on the element g=3, n times.
For a negative exponent, one must add the inverse of g=3 with respect to addition, thus g-1=-3.
$$ 3^{-4} = (-3)+(-3)+(-3)+(-3) = -12 $$
If the exponent is zero, the outcome is simply the neutral (identity) element of the operation, which for addition, is zero.
$$ 3^0 = 0 $$
This illustrates that <3> encompasses all integer multiples of 3 within the set of integers, making it a subgroup of (Z,+).
$$ <3> = \{ 3Z \} \subseteq \{ Z \} $$
As such, it is an infinite cyclic subgroup, since there are an infinite number of both positive and negative multiples of 3.
And so forth.
