The Order of an Element of a Group
If $$G$$ is a group and $$a$$ is an element of group $$G$$, the order (or period) of $$a$$ is the least positive integer $$n$$, such that
\[{a^n} = e\]
If there exists no such integer, we say that $$a$$ is a finite order or zero order. We shall use the notation $$O\left( a \right)$$ for the order of $$a$$.
Note that the only element of order one in a group is the identity element $$e$$.
Important Note: If there exists a positive integer $$m$$ such that $${a^m} = e$$, then the order of $$a$$ is definitely finite. Also we must have $$O\left( a \right) \leqslant m$$. When $${a^m} = e$$, then the question of order of a being greater than $$m$$ does not arise. At the most it can be equal to $$m$$. If $$m$$ itself is the least positive such that $${a^m} = e$$, then we will have $$O\left( a \right) = m$$.
Example:
Find the order of each element of the multiplicative group $$G$$, where $$G = \left\{ {1, – 1,i, – i} \right\}$$
Since 1 is the identity element, its order is 1.
Now
\[{\left( { – 1} \right)^1} = – 1,\,\,{\left( { – 1} \right)^2} = \left( { – 1} \right)\left( { – 1} \right) = 1\]
Hence the order of -1 is 2.
Again
\[{i^1} = i,\,\,\,{i^2} = – 1,\,\,\,{i^3} = – i,\,\,\,{i^4} = 1\]
Therefore the order of $$i$$ is 4.
Similarly,
\[{\left( { – i} \right)^1} = – i,\,\,\,{\left( { – i} \right)^2} = {i^2} = – 1\]
\[{\left( { – i} \right)^3} = i,\,\,\,{\left( { – i} \right)^4} = 1\]
Hence the order of -1 is 4.
Mukund Karki
February 3 @ 8:50 am
How did i^2 = -1 couldn’t find the explanation throughout the internet
T
March 24 @ 4:26 pm
Here “i” refers to the “imaginary numbers” which are defined by i^2=-1. This is the basis of complex numbers, which have a real and an imaginary part, like a+bi, where the real part is RE(a+bi)=a, and the imaginary part is IM(a+bi)=bi.
These numbers are really important in a lot of fields of Mathematics like Field theory.
If this doesn’t make sense just yet, the important thing to take away is that i^2=-1 is the definition of the imaginary number “i”
Basant prasad
August 23 @ 8:14 am
Order of element(-1)=2, not 4