# Group having 3 proper subgroups

• May 27th 2010, 03:39 AM
Chandru1
Group having 3 proper subgroups
What can we say about a group which has exactly 3 proper subgroups?
• May 27th 2010, 04:05 AM
tonio
Quote:

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
• May 27th 2010, 04:12 AM
Swlabr
Quote:

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, $H \lneq G$. So we are looking for a group with precisely two non-trivial subgroups (although I think your hint is still the way to go).
• May 27th 2010, 04:29 AM
TheArtofSymmetry
Quote:

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 $p^3$ can be one of examples, where p is a prime number.

See my previous post here .
• May 27th 2010, 04:40 AM
Swlabr
Quote:

Originally Posted by TheArtofSymmetry
As tonio said, a cyclic group of order $p^3$ 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 $V_4$, the klein 4-group (or, more generally, the cross-product of two groups of prime order).
• May 27th 2010, 05:28 AM
tonio
Quote:

Originally Posted by Swlabr
No - proper means it is properly contained in it. That is to say, $H \lneq G$. 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