For cyclic groups we can give more detailed information on the order of its elements:
Proposition
Let G be a cyclic group of order n with generator g. Then
- every subgroup of G is cyclic;
- <gk> = <gd> for d = gcd(n,k); it is a subgroup of order n/d;
- gk generates G if and only if gcd(k,n) = 1.
The above implies that the number of generators
in a cyclic group of order n equals
(n). (Here
denotes the Euler indicator.)
We can characterize the cyclic groups by the number of elements of a given order. This characterization will be of use in the next chapter.
Characterization of
finite cyclic groups
Let G be a group of order n. Then the following three statements are equivalent:
- G is cyclic.
- For every divisor d of n there are exactly
(d) elements
of order d in G.
- For every divisor d of n there are exactly d elements x in G with xd = e.