# Direct product and Decomposition of a symmetric group

• Dec 10th 2008, 10:03 PM
aliceinwonderland
Direct product and Decomposition of a symmetric group
I am looking for a mechanism to find a decomposition of symmetric groups. For finitely generated abelian group G, there is a mechanism to decompose G such that G is isomorphic to a direct sum of cyclic groups.
For symmetric groups, it seems a bit complex for me to find it.
For example,

(1) is it possible for $\displaystyle S_{3}, S_{4}, S_{5}$ to be decomposed as a direct product of groups?
In more general cases, is there any mechanism to decompose $\displaystyle S_{n}$ as a direct product of groups?

(2) is there any mechanism to find an isomorphic group of a direct product of symmetric groups? Let's say,
$\displaystyle G = S_{3} \times S_{4} \times S_{5}$
Any isomorphic group of above G?
• Dec 11th 2008, 05:30 AM
Gamma
I can give you a cheesy isomophism
recall the dihedral group of order 6, $\displaystyle D_6 \cong S_3$ in fact every non abelian group of order 6 is isomorphic. So $\displaystyle D_6 \times S_4 \times S_5 \cong S_3 \times S_4 \times S_5$