Hey there guys.

Let G be a group of order 12. Show by a Sylow counting argument that if G does not have a normal subgroup of order 3 then it must have a normal subgroup of order 4.

Deduce that G has one of the following forms:

(i) $\displaystyle C_3 \rtimes C_4$

(ii) $\displaystyle C_3 \rtimes (C_2 \times C_2)$

(iii) $\displaystyle C_4 \rtimes C_3$ or

(iv) $\displaystyle (C_2 \times C_2) \rtimes C_3$

Hence, classify all groups of order 12 up to isomorphism.

Any suggestions on how to go about doing this one please?

I will attempt myself once I have a good idea of what to do . Thnx