Prime order implies no proper nontrivial subgroup

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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.