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.
Or:
If , then you would be able to find two subgroups of such that:
1) is isomorphic to , is isomorphic to
2) is a normal subgroup of , is a normal subgroup of
3)
4) .
And this is clearly impossible since only nontrivial normal subgroup of S3 is the one generated by 3-cycle.