Differences between Groups

[jacobsldr]

After earning a 4.0 thru the Calc series, I am now being humbled by the subject that is abstract algebra, any help would be appreciated.

Given assignment to research groups of order 12, bring back a paper describing each group in detail.

I was able to find 5 groups of order 12. 2 Abelian (Cyclic group 12, Finite Group C2xC6) and 3 non Abelian (A4 alternating group degree 4, Dih6 Dihedral group, and Dic3 Dicyclic group)

I am stuck on identifying the structural property that would distinguish these groups from each other. They are all order 12 so cardinality it same. I know that communtativity separates the abelian from non-abelian groups, but what property distinguishes the 2 abelians from each other? I see that A4 elements have order of 4 or less and Dic3 has a subgroup with 6 elements, is this considered a structural property difference?

Thanks for any info that can help my understand this better,

jacob.....who really thought he was a smart math guy....until abstract algebra, sigh.

Edit: Upon further thinking, I believe the finite group C2XC6 is not cyclic, which would make it structurally different from C12, is this right???

 

[daon]

Cyclic 12 has 4 generators:{1,5,7.11} i.e. there are 4 elts. of order 12
C2xC6 has none. All have order <=6.

edit: remember that GxF is cyclic iff G and F are cyclic and gcf(|G|, |F|)=1.

 

[jacobsldr]

Thanks so much!!

That got the ball rolling for me. By looking closer at each of the subgroups and their orders and using the differences of the centers of the groups I was able to now find at least one structural difference for each of the 5 groups of order 12.

And thanks for the added edit. I was trying to grasp in my mind an easy way to tell if the group product was cyclic or not, and that one line I can commit to memory as a quick check.

jacob....i will learn this, i will learn this, i will learn this....