Let #G# is cyclic group and #|G|=48#. How do you find all of subgroup of #G# ?
1 Answer
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 than#1# generator.- The order of
#H# is a factor of#N# .
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 >#