Can anyone prove that there are only 2 non-isomorphic groups of order 6.

Printable View

- December 16th 2009, 07:12 PMpleasehelpme1Groups of Order 6
Can anyone prove that there are only 2 non-isomorphic groups of order 6.

- December 16th 2009, 07:49 PMDrexel28
- December 16th 2009, 08:05 PMNonCommAlg
it's a nicer problem to prove that for any prime number there are only two groups (up to isomorphism) of order (Nod)

- December 17th 2009, 04:56 AMpleasehelpme1...
can your prove that then.

that for any prime number http://www.mathhelpforum.com/math-he...97a8c47a-1.gif there are only two groups (up to isomorphism) of order http://www.mathhelpforum.com/math-he...7c8b6f69-1.gif - December 18th 2009, 11:39 AMDinkydoe
Every group has generators:

You can start by showing that G is either

1.generated by an element of order 6

2.an element of order 3 and an element of order 2.

Since all orders of the generators divide the group-order

(This follows from the Lagrange Theorem: #[G/H].#[H] = #G

For any subgroup H)

Once you figured that out you show that (1) and (2) are not isomorphic.

Secondly you show that 2 groups G and G' of type (1) are isomorhpic

and 2 groups of type (2) are isomorphic.

(This isomorphism is made by sending generators to generators) - December 18th 2009, 11:49 AMSwlabr
- December 18th 2009, 12:17 PMNonCommAlg
suppose G is a group of order 6 and let be with since we have now consider two cases:

1) : in this case and thus where is the cyclic group of order 6.

2) : since we have for some if then which is not true. if then which is

the first case. so and therefore the dihedral group of order 6.