# Quick verification of something...

• Dec 15th 2009, 04:23 PM
ElieWiesel
Quick verification of something...
Is this true?
H, N, G are groups

$H \triangleleft N \triangleleft G$ implies $H \triangleleft G$
• Dec 15th 2009, 05:02 PM
tonio
Quote:

Originally Posted by ElieWiesel
Is this true?
H, N, G are groups

$H \triangleleft N \triangleleft G$ implies $H \triangleleft G$

It is false: $\{(1),(12)(34)\}\,\triangleleft\, \{(1),(12)(34),(13)(24),(14)(23)\}\,\triangleleft\ , A_4\,,\,\,but\,\,\,\{(1),(12)(34)\}\,\ntrianglelef t\, A_4$

Tonio
• Dec 15th 2009, 05:39 PM
ElieWiesel
What if N is cyclic? Does it then imply that H is abelian, and hence H a subgroup of G is normal in G?
• Dec 15th 2009, 06:58 PM
tonio
Quote:

Originally Posted by ElieWiesel
What if N is cyclic? Does it then imply that H is abelian, and hence H a subgroup of G is normal in G?

Yes, then H is normal in G, but not your "hence", because if N, H are abelian then still it isn't true that H must be normal, as my counterexample shows. I leave it to you to prove that if N is cyclic then H is normal in G.

Tonio