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


1. The length of the series is unique for a group.
2. Two non-isomorphic groups cannot have isomorphic series.


Thanks. (My apologies if this turns out a trivial question.)



The X-series (for groups)
This is called a [chief series](https://en.wikipedia.org/wiki/Chief_series) or principal series. The quotients of the terms in the series are called chief factors, and the length of the series is called the chief length. Non-isomorphic groups can have isomorphic chief factors. It depends on your definition of isomorphism of chief series whether non-isomorphic groups can have isomorphic chief series (for instance, if G itself is part of the data of the chief series, then G is obviously determined by its series).


The chief length is a group isomorphism invariant. There is a version of Schreier's refinement theorem that applies to chief series. Groups can have many chief series, though some groups have only one, like S4 and SL(2,3) (which have the same chief factors but in a different sort of order).


