1. ## groups

Is the symmetric group S[/SIZE]3 a direct product of nontrivial groups?

could anyone help me ?? please?
(what does it mean by 'direct product of nontrivial groups'?)

2. Originally Posted by jin_nzzang
[snip]
(what does it mean by 'direct product of nontrivial groups'?)
A "trivial group" is a group with one element.

Definition:Internal Direct Product - ProofWiki
may help to remind you what a "direct product" is.

3. Originally Posted by jin_nzzang
Is the symmetric group S[/size]3 a direct product of nontrivial groups?
[snip]
No. Else it would only be the direct product of groups of order 2,3. but any groups with order 2 is isomorphic to $Z_2$ and any groups with order 3 is isomorphic to $Z_3$. That means $S_3\cong Z_2\times Z_3\cong Z_6$,which is cyclic and clearly impossible.

4. Or:

If $S_3 = H' \times K'$, then you would be able to find two subgroups $H, K$ of $S_3$ such that:
1) $H$ is isomorphic to $H'$, $K$ is isomorphic to $K'$
2) $H$ is a normal subgroup of $S_3$, $K$ is a normal subgroup of $S_3$
3) $HK = S_3$
4) $H \cap K = 0$.

And this is clearly impossible since only nontrivial normal subgroup of S3 is the one generated by 3-cycle.