MHB Composition Series for Group Order 30

[Pratibha]

How to find the composition series of a group of order 30

 

[Fallen Angel]

Hi,

It may depend on the caharacteristics of the group.

If your group it's abelian then use the Abelian's groups structure theorem.

If not, I think we need to know something more.

 

[Pratibha]

what is statement of abelian structure theorem?

 

[johng1]

This may be beyond your knowledge, but here goes:
A group of order 30 has a normal subgroup of index 2 (Any group of even order with a cyclic Sylow 2 subgroup has a normal subgroup of index 2; proved by considering the Cayley representation of G as a permutation group.) So let H be normal of order 15. By the Sylow theorems H has a normal Sylow 5 subgroup K (cyclic) and a normal Sylow 3 subgoup L. So one composition series (actually a chief series--all subgroups in series are normal in G) is:
$$<1>\,\trianglelefteq L\trianglelefteq H\trianglelefteq G$$ with the factors cyclic of orders 2, 3 and 5. The Jordan-Holder theorem says all composition series have the same factors.

 

[Fallen Angel]

Every finitely generated abelian group $G$ is isomorphic to a direct product

$\Bbb{Z}^{n}\oplus \Bbb{Z}_{a_{1}}\oplus \ldots \oplus \Bbb{Z}_{a_{k}}$

Moreover, we can order it in such a way that $a_{i}$ divides $a_{i+1}$ for every $i=1,\ldots, k-1$

PS: It's easier than I thought using the way johng has just posted