g-e in augmentation ideal of subgroup H iff g in H

Question

Let $H$ be a subgroup of $G$ and $J_H$ the augmentation ideal of $H$. Then for $g\in G$ we have $g-e \in J_H$ iff $g\in H.$

In Daniel E. Cohen's Groups of Cohomological Dimension One, which can be found here: http://www.springer.com/us/book/9783540057598 (Specifically Lemma 4.1 which is available for preview.)

This fact is proven, but I do not understand this proof (of part (i)). Is there someone who can explain this, or give more references?

Answer 1

I cannot see the preview of the proof, but the result seems pretty straightforward. It is a known fact that the additive group of the augmentation ideal is (freely) generated by elements of the form {g-e}, for g non trivial elements of G(I don't know how to tex here, so please bear with me).

So, if g is an element of H, then g-e lies inside J_H.

For the reverse, if g-e lies inside J_H, then g-e is a Z-combination of elements of H and thus, it lies inside H. Hope I helped (and not got too sloppy XD).