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'?)
A "trivial group" is a group with one element.
This page:
Definition:Internal Direct Product - ProofWiki
may help to remind you what a "direct product" is.
No. Else it would only be the direct product of groups of order 2,3. but any groups with order 2 is isomorphic to $\displaystyle Z_2$ and any groups with order 3 is isomorphic to $\displaystyle Z_3$. That means $\displaystyle S_3\cong Z_2\times Z_3\cong Z_6$,which is cyclic and clearly impossible.
Or:
If $\displaystyle S_3 = H' \times K'$, then you would be able to find two subgroups $\displaystyle H, K$ of $\displaystyle S_3$ such that:
1) $\displaystyle H$ is isomorphic to $\displaystyle H'$, $\displaystyle K$ is isomorphic to $\displaystyle K'$
2) $\displaystyle H$ is a normal subgroup of $\displaystyle S_3$, $\displaystyle K$ is a normal subgroup of $\displaystyle S_3$
3) $\displaystyle HK = S_3$
4) $\displaystyle H \cap K = 0$.
And this is clearly impossible since only nontrivial normal subgroup of S3 is the one generated by 3-cycle.