# Thread: [SOLVED] Non-Commutative Groups

1. ## [SOLVED] Non-Commutative Groups

Just a quick curiosity question, since I spent the afternoon building group multiplication tables. (I'm reviewing a bit for a new book I bought.)

What is the smallest group that is non-commutative? I'm driving myself out of my mind trying to construct small (that is to say sets of less than 8 elements) non-commutative groups.

Thanks!
-Dan

2. Originally Posted by topsquark
Just a quick curiosity question, since I spent the afternoon building group multiplication tables. (I'm reviewing a bit for a new book I bought.)

What is the smallest group that is non-commutative? I'm driving myself out of my mind trying to construct small (that is to say sets of less than 8 elements) non-commutative groups.

Thanks!
-Dan
The symettric group on {1,2,3}. It has 3! = 6 elements.

3. Originally Posted by ThePerfectHacker
The symettric group on {1,2,3}. It has 3! = 6 elements.
A non-commutative symmetric group? Man I really DO need a review. I was thinking symmetric groups had to be commutative.

Well, at least I have a good place to start tomorrow morning!

Thanks.
-Dan

4. Originally Posted by topsquark
I was thinking symmetric groups had to be commutative.
I wish! Then all of group theory would be so easy.