- #1

mnb96

- 715

- 5

if we consider a group

*G*and two subgroups

*H,K*such that [itex]HK \cong H \times K[/itex], then it is possible to prove that:

[tex]G/(H\times K) \cong (G/H)/K[/tex]

Can we generalize the above equation to the case where [itex]HK \cong H \rtimes K[/itex] is the

*semidirect product*of

*H*and

*K*?

Clearly, if

*HK*is a semidirect product, then it might not be normal in

*G,*so my guess is that the best we can do is to calculate the quotient [itex]G / \langle H \rtimes K \rangle^G[/itex] where [itex]\langle H \rtimes K \rangle^G[/itex] denotes the

*normal closure (or conjugate closure)*in

*G*of the semidirect product.

Do you have any hint on how to do this?