For finite groups, each element \(e\) has:
\(e^n=I\)
For some \(n\in \mathbb{N}\)
Where \(I\) is the identity element.
The order of the group is the smallest value of \(n\) such that that holds for all elements.
For example in the multiplicative group \(G=\{-1,1\}\) the order is \(2\).
Or:
\(|G|=2\)
Additionally
\(|-1|=2\)
\(|1|=1\)
Consider the set of natural numbers and addition modulo 4. This forms a group containing:
\(\{0,1,2,3\}\)
This can be written as \(Z_4\) or more generally as \(Z_n\), or \(Z/nZ\).
The trivial group is the group with just the identity member \(I\).
Consider the multiplicative group of \(< i >\).
This contains \(\{1,-1,i,-i \}\).
This is also automorphic to the natural number and modulo addition group above.
We can define finite cyclic groups of size \(n\) using the generating element \(z^{\dfrac{1}{n}}\). This is isomorphic to the general cyclic group \(C_n\), and to \(Z/nZ\).
We can generate a group with a single element, it is a cyclic group.
For example, we can define a group \(G=<1>\) which gives us the additive group of integers.
We can define a group through a generating set and an operation.
And define the group as \(G=< S >\)