# How does a short exact sequence say something about a group?

I have a follow-up question to my question here: How are simple groups the building blocks?

In that question I asked about what it means when we say that the simple (finite) groups are the building blocks of all finite groups. It seems it all comes down to understanding that if you have a short exact sequence: $$1 \to H \to G \to G / H \to 1$$ then $G$ us built out of $H$ and $G /H$. From a comment to an answer the point is made that that there are various ways that one can say that $G$ is built out of $H$ and $G/H$.

My follow-up question is what this means? What are examples of what can be said? How can, for example, can information about $G$ be "lifted" from $H$ and $G/H$?

Some group-theoretic properties can be lifted from $H$ and $G/H$ to $G$. One examples is solubility.
Others can not. Examples are being abelian, or nilpotent. (For both, take $G$ to be $S_3$, and $H = A_3$.)
• One can prove that if $H$ and $G/H$ are soluble, then so is $G$. See for instance Theorem 4.1.2(b) on page 82 here: maths.qmul.ac.uk/~pjc/notes/gt.pdf – Andreas Caranti Feb 18 '16 at 19:56
Indeed, there are various ways that one can say that $G$ is built out of $H$ and $G/H$. What this means ? Perhaps a nice example is to take $H$ and $Q=G/H$ as the cyclic group $C_p$ for a prime $p$. It is well known that there are exactly $p$ inequivalent extensions of $C_p$ by $C_p$, e.g.,short exact sequences $$1\rightarrow C_p\rightarrow G \rightarrow C_p\rightarrow 1,$$ and we know that $G$ must have $p^2$ elements, i.e., must be an abelian group, hence either $C_p\times C_p$, or $C_{p^2}$. The equivalence classes of such extensions are measured by the second cohomology group $$H^2(C_p,C_p)\cong C_p.$$ The question which algebraic properties of groups are preserved under extension is quite difficult in general. Solvability is an easy example, but there are more complicated ones, e.g., strong embeddability is preserved under group extensions, or being linear algebraic $k$-groups is preserved, too.