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.

Printable View

- Dec 2nd 2013, 03:49 PMStormeyHelp with a proof - the commutator subgroup
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. - Dec 2nd 2013, 04:07 PMPlatoRe: Help with a proof - the commutator subgroup
- Dec 2nd 2013, 04:52 PMjohngRe: Help with a proof - the commutator subgroup
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. - Dec 3rd 2013, 03:55 AMStormeyRe: Help with a proof - the commutator subgroup
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.