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

Statement

If a finite group has order $p$ for some prime number $p$, 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.

Facts used

1. Lagrange's theorem: In the simplistic form here, the order of any subgroup of a group divides the order of the group.

Proof

Given: A group $G$ of order equal to a prime number $p$. A subgroup $H$ of $G$.

To prove: $H$ is either equal to $G$ or the trivial subgroup.

Proof: By fact (1), the order of $H$ divides the order of $G$, which is the prime number $p$. Thus, $H$ has order equal to $1$ or $p$. We consider both cases:

• If $H$ has order $1$, it must be the trivial subgroup, because $H$ must contain the identity element.
• If $H$ has order $p$, it has to equal the whole group $G$.