Creating noncyclic groups of certain order

[Entropee]

How would I construct noncyclic groups of whatever order I want? For example g is order 8.

 

[Office_Shredder]

If you want an even order the dihedral group is a good example... for odd I don't really know how to do it. Based on the wikipedia article

http://en.wikipedia.org/wiki/List_of_small_groups

in which only a single noncyclic odd group is given, Z₃xZ₃, there apparently aren't many of them.

 

[lavinia]

How would I construct noncyclic groups of whatever order I want? For example g is order 8.

a group of prime order must be cyclic

if the order is nxm can you think of an example?

 

[Entropee]

We'll in some cases I can but for example if the order is 25 I can't think of one that is NON cyclic, otherwise it would be easy.

 

[pasmith]

We'll in some cases I can but for example if the order is 25 I can't think of one that is NON cyclic, otherwise it would be easy.

$C_5 \times C_5$ is an abelian group of order 25 in which every element other than the identity is of order 5.

($C_n \times C_m$ is not isomorphic to $C_{n+m}$ unless $n$ and $m$ are coprime.)

 

[lavinia]

here is a group of order 6.

b$^{3}$ = a$^{2}$ = id

aba$^{-1}$ = b$^{2}$

Try to show that every group of order 15 is abelian.