# isomorphisms

• October 29th 2008, 12:59 PM
morganfor
isomorphisms
How does one show that a G is isomorphic to the external direct product of H and K if G=HK and if H and K have unique elements.
Also how does one apply this to proving U(15) is isomorphic to the external direct product of U(3) and U(5) and how would you show if U(15) is cyclic or not?
• October 30th 2008, 09:10 AM
ThePerfectHacker
Quote:

Originally Posted by morganfor
How does one show that a G is isomorphic to the external direct product of H and K if G=HK and if H and K have unique elements.
Also how does one apply this to proving U(15) is isomorphic to the external direct product of U(3) and U(5) and how would you show if U(15) is cyclic or not?

Are you asking to prove if $H,K \triangleleft G$ with $G = HK$ and $H\cap K = \{ e \}$ then $G\simeq H\times K$?
• October 30th 2008, 07:20 PM
morganfor
yes but H and K are not normal to G