I need to prove the following:
Let G be a group.
If $\displaystyle H<G$ s.t. $\displaystyle S<H$, then $\displaystyle H\triangleleft G$ (S is the commutator subgroup).
I just need a lead here...
Thank you in advance.
I need to prove the following:
Let G be a group.
If $\displaystyle H<G$ s.t. $\displaystyle S<H$, then $\displaystyle H\triangleleft G$ (S is the commutator subgroup).
I just need a lead here...
Thank you in advance.
Stormey,
I think the commutator subgroup of G is universally denoted by G'. The easiest way to answer your question is to work in the factor group G/G'. However a direct approach is:
Let h be in H and g in G. Then [g,h^{-1}]=g^{-1}hgh^{-1} is in H and so g^{-1}hg is in H.
Sorry.
here it is again with standard notations:
I need to prove the following:
Let G be a group.
If $\displaystyle H\leq G$ s.t. $\displaystyle H'\leq H$, then $\displaystyle H\triangleleft G$.
I just need a lead here...
Thank you in advance.
Thanks for the help.