Prime order implies no proper nontrivial subgroup
This fact is an application of the following pivotal fact/result/idea: Lagrange's theorem
View other applications of Lagrange's theorem OR Read a survey article on applying Lagrange's theorem
If a finite group has order for some prime number , then it has no proper nontrivial subgroup. In other words, the only possible subgroups of the group are the trivial subgroup and the whole group.
- Lagrange's theorem: In the simplistic form here, the order of any subgroup of a group divides the order of the group.
Given: A group of order equal to a prime number . A subgroup of .
To prove: is either equal to or the trivial subgroup.
Proof: By fact (1), the order of divides the order of , which is the prime number . Thus, has order equal to or . We consider both cases:
- If has order , it must be the trivial subgroup, because must contain the identity element.
- If has order , it has to equal the whole group .