Question

How do you prove that if a group G has no non-trivial subgroups, then it is finite of prime order?

Answer 1

In the above question, we are given a group G. Also, it is given that the group G has no non-trivial subgroups. This means all the subgroups are trivial, so is a group consisting of a single element. The single element of the trivial group is the identity element. We have to prove that the given group G is finite and has its order as a prime number. A subgroup of a group G is defined as a subset of G that forms a group with the same law of composition. For example, the positive natural numbers form a subgroup of the group of integers with group law of addition.

Complete step by step answer:
Given that, a group G which has no non-trivial subgroups.
We have to prove that the group G is finite and has a prime order.
Every group G contains two trivial or improper subgroups, G itself and the group consisting of the identity element alone, i.e. \left\\{ i \right\\} .
All the other subgroups are called non-trivial or proper subgroups.
Hence, any group G has at least two subgroups, they are the trivial subgroups.
Now, it is given that the group G has no non-trivial subgroups.
For G to have a finite order, G must be a finite group.
That means the group G has only trivial subgroups, which are \left\\{ i \right\\} and the group G itself.
Now, from Lagrange's theorem, the order of any subgroup of G must divide the order of the group G.
Let the order of the group G be $p$ , then the order of the subgroups of G are $p$ and $1$ , as the order of \left\\{ i \right\\} is $1$ .
Hence, the order of the group G, that is $p$ , has only $2$ factors which are $p$ and $1$ .
Therefore, we conclude $p$ is prime and G is a finite group of prime order.

Note:
In mathematics, a group is a set equipped with an operation that combines any two elements to form a third element while holding the law of associativity as well as having the identity element and inverse elements. These above given three conditions are called group axioms and they hold for number systems and many other mathematical structures. For example, all the integers together with the multiplication or addition operation form a group.