Are all groups of order 6 cyclic? For exactly the value of n=6, it is true that every group of order n is cyclic? I believe it is false, but cannot find the contradiction!
 A: Up to isomorphism, there are exactly two groups of order $6$: 


*

*the cyclic group of order $6$: $\mathbb Z_6$, or if you prefer, $C_6$,

*the non-abelian symmetric group $S_3$
Put differently, every group of order $6$ is isomorphic to exactly one of two groups:


*

*$\mathbb Z_6$ 

*$S_3$


For example, as pointed out in the comments, the dihedral group of order $6$ (the group of symmetries of the equilateral triangle) is isomorphic to $S_3$.

N.B. Knowing that there are exactly two groups, up to isomorphism, of order $6$ is a good thing to know, just as is the fact that there are exactly two groups, up to isomorphism, of order $4$: $\mathbb Z_4$: the cyclic group of order $4$, and the Klein 4-group, which is abelian, but not cyclic.
A: $S_3$ is of order 6, and nonabelian.
A: Every abelian group $G$ of order 6 is cyclic.
If $a$ is an element of order 2, $a^2=1$ and $b$ is an element of order 3 $b^3=1$ what is the order of the element $ab$?
The existence of elements $a$ and $b$ follows from the fact than for every prime divisor of the order of a group there exists an element of such order (Cauchy)
In general if  $G$ isn't abelian and has order $|G|=pq$,  $p,q$ primes then
$G=<a,b|b^p=1,a^q=1,a^{-1}ba=b^r>$ where $r\not\equiv 1 \mod p$ and $r^q\equiv 1 \mod p$ and $q|p-1$.
