# Math Help - groups

1. ## groups

Classify groups of order 6 by analyzing the following three cases.
(a) G contains an element of order 6
(b) G contains an element of order 3 but none of order 6
(c) All elements of G have order 1 or 2

could anyone please help me with this problem ?

2. a) if an element has order 6, in a group of order 6, it generates the whole group, i.e. this is $\mathbb{Z}_6$
b) This would be the lone nonabelian group of order 6. $S_3$ or $D_6$ they are the same.
c) I don't think there is such a group, the sylow 3 subgroup is normal and they have trivial intersection so HK is a subgroup of order 6, so G must be a semi direct product, ie one of the two i mentioned before. Alternatively, cauchy's theorem guarantees the existence of an element of order 3.