# Math Help - Sylow's Counting Argument

1. ## Sylow's Counting Argument

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) $C_3 \rtimes C_4$
(ii) $C_3 \rtimes (C_2 \times C_2)$
(iii) $C_4 \rtimes C_3$ or
(iv) $(C_2 \times C_2) \rtimes C_3$

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