Let G is cyclic group and |G|=48. How do you find all of subgroup of G ?
1 Answer
Feb 23, 2018
The subgroups are all cyclic, with orders dividing
Explanation:
All subgroups of a cyclic group are themselves cyclic, with orders which are divisors of the order of the group.
To see why, suppose
If
So
In particular, if
Also not that if
We can deduce:
H has no more than1 generator.- The order of
H is a factor ofN .
In our example
C_1 ,C_2 ,C_3 ,C_4 ,C_6 ,C_8 ,C_12 ,C_16 ,C_24 ,C_48
being:
< > ,< a^24 > ,< a^16 > ,< a^12 > ,< a^8 > ,< a^6 > ,< a^4 > ,< a^3 > ,< a^2 > ,< a >