Thread: Help with a proof - the commutator subgroup

1. Help 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...

2. Re: Help with a proof - the commutator subgroup

Originally Posted by Stormey
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).
You really should define all the terms and symbols in your question.
Why do you think the notation is standard?

3. Re: 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-1hgh-1 is in H and so g-1hg is in H.

4. Re: Help with a proof - the commutator subgroup

Originally Posted by Plato
You really should define all the terms and symbols in your question.
Why do you think the notation is standard?
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...