It goes without saying that the name in the title tentatively refers to a series whose name one does not know yet and probably in the future I may come with a post titled "The x-sequence" or "The x-function" ,etc. I do not know whether the quotient series I am going to construct is known as a "descending quotient series". Okay, it's enough.
Let $G$ be a non-abelian non-simple finite group. Let $H_0$ be one of the smallest proper normal subgroups of $G$. Now if $G/H_0$ is non-simple, call it $G_1$ and let $H_1$ be one of the smallest normal subgroups of $G_1$. Next if $G_1/H_1$ is non simple, call it $G_2$ and let $\ldots$ Continue the process until a simple group is found (which is always possible). I was wondering whether
The length of the series is unique for a group.
Two non-isomorphic groups cannot have isomorphic series.
Thanks. (My apologies if this turns out a trivial question.)