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