# Thread: Group having 3 proper subgroups

1. ## Group having 3 proper subgroups

What can we say about a group which has exactly 3 proper subgroups?

2. Originally Posted by Chandru1
What can we say about a group which has exactly 3 proper subgroups?

Since any non-trivial group always has two trivial subgroups (the trivial one and the whole group), we're looking for a group with one single non-trivial subgroup...hint: how many primes can divide the group's order?

Tonio

3. Originally Posted by tonio
Since any non-trivial group always has two trivial subgroups (the trivial one and the whole group), we're looking for a group with one single non-trivial subgroup...hint: how many primes can divide the group's order?

Tonio
No - proper means it is properly contained in it. That is to say, $\displaystyle H \lneq G$. So we are looking for a group with precisely two non-trivial subgroups (although I think your hint is still the way to go).

4. Originally Posted by Chandru1
What can we say about a group which has exactly 3 proper subgroups?
As tonio said, a cyclic group of order $\displaystyle p^3$ can be one of examples, where p is a prime number.

See my previous post here .

5. Originally Posted by TheArtofSymmetry
As tonio said, a cyclic group of order $\displaystyle p^3$ can be one of examples, where p is a prime number.

See my previous post here .
As a warning to the OP, this does not classify them all. Another example would be $\displaystyle V_4$, the klein 4-group (or, more generally, the cross-product of two groups of prime order).

6. Originally Posted by Swlabr
No - proper means it is properly contained in it. That is to say, $\displaystyle H \lneq G$. So we are looking for a group with precisely two non-trivial subgroups (although I think your hint is still the way to go).

Yes. I oversaw the word "proper" in the OP, but still the hint remains...though nevertheless there are OTHER examples: for example, a cyclic group of order 10...

Tonio