Help with a proof - the commutator subgroup

• Dec 2nd 2013, 03:49 PM
Stormey
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...

• Dec 2nd 2013, 04:07 PM
Plato
Re: Help with a proof - the commutator subgroup
Quote:

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?
• Dec 2nd 2013, 04:52 PM
johng
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.
• Dec 3rd 2013, 03:55 AM
Stormey
Re: Help with a proof - the commutator subgroup
Quote:

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...

Quote:

Originally Posted by johng
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.

Thanks for the help.