Let $FG$ be a finite group ring of a finite non abelian $p$-group $G$ over finite field $F.$ It is well known that augmentation ideal $\Delta(G)=J(FG)$ has basis as the set $\{g-1:g\in G, g\ne 1\}$, being kernel of the augmentation map $f:FG\rightarrow F.$ Now my question is what is a generating set of $1+\Delta(G)?$ Can i say that its generating set is the set $\{g\in G:g\ne 1\}?$
My real question is that as i proved that $1+\Delta(G)$ is a finite non-abelian group. Can i say that exponent of this group will not exceed exponent of the group $G?$ Is there any way to find cardinality of $1+\Delta(G)?$ Please help me . Thanks .