Prologue~
“A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas. The mathematician’s patterns, like the painter’s or the poet’s, must be beautiful; the ideas, like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics.”
— From A Mathematician’s Apology by G. H. Hardy
Verse I: The Problem~
An extension problem in group theory typically deals with construction of new groups with the already existing ones.
What that asks of you is to imagine two groups- $N$ and $Q$, and then search the group(s) (namely $G$) that would consist of $N$ at its core (as a normal subgroup) and $Q$ as a visible skeleton (as a quotient).
This problem can look something like:$$1 \rightarrow N \rightarrow G \rightarrow Q \rightarrow 1 $$
So, can we construct $G$ knowing only $Q$ and $N$?
Well, there is more to a sequence like that!
Verse II: The Rise~
Hurewicz introduced the idea of exact sequences in 1941 (without naming it) to describe boundary maps in cohomology. Kelley and Pitcher coined the term "exact sequence" in 1947, acknowledging its use by Eilenberg and Steenrod in earlier courses.
The variations of the story surrounding the naming of the term, 'exact sequences' can be accessed here.
It seems rather funny to me that the term 'right' was actually 'exact'!
Another variation can be accessed here.
Verse III: The sequence~
A split extension is an extension $1 \rightarrow K \rightarrow G \rightarrow H \rightarrow 1$ with a homomorphism $\phi : H \rightarrow G$, if the short exact sequence induces the identity map on $H$ i.e., $p \circ \phi = id_H$ where $p: G \rightarrow H$ is the projection . In this situation, it is usually said that $\phi$ splits the above exact sequence.
Verse IV: The Way~
So to tackle the problem, one approach could be (will be) to construct the group using direct products of $N$ and $Q$ (a trivial case, perhaps) and every element of $G$ is then simply represented as an ordered pair, where one element comes from $N$ and the other from $Q$. In such a case, we find $G$ to contain a copy of $N$ as well as $Q$!
$$G \cong N \times Q$$ And, nothing mystical really happens— instead the two groups coexist peacefully without entanglements (in a trivial sense, that is).
But are all cases trivial? Certainly no!
Consider a semidirect product, $$G \cong N \rtimes_\phi Q$$ Here, $Q$ no longer simply co-inhabits $G$ with $N$, instead it acts upon the automorphism group of $N$, and the fabric of our beloved group $G$ here, becomes richer in tapestry.
Is that where our thinking stops? Certainly no (again)!
There exist cases where $G$ cannot be expressed as a direct or a semidirect product of $N$ and $Q$ and those give rise to the non-split extensions. Which implies that even though $N$ and $Q$ are present in the groups, one cannot unseam the stitches of the fabric and obtain copies of them.
A small example would be to consider $N= \mathbb{Z}/ 2 \mathbb{Z}$ and $Q= \mathbb{Z}/ 2 \mathbb{Z}$. Now, constructing it's direct product, we obtain $\mathbb{Z}/ 2 \mathbb{Z} \times \mathbb{Z}/ 2 \mathbb{Z}$ which is the split extension and surprisingly enough, the semi direct product also leads us to the same group, which (voilà!) is isomorphic to the Klein 4 group.
Verse V: The other Way~
But, there also exists a possible non-split extension- $G \cong \mathbb{Z} / 4 \mathbb{Z}$.
Then how do we know which one is our $G$? Because $\mathbb{Z} / 4 \mathbb{Z}$ is a non-split extension and hence cannot be written as a semidirect product.
Well, this ambiguity, with a small example underlines our extension problem which arises from this multiplicity of possibilities.
Verse VI: The 'exact'~
An exact sequence is a sequence of morphisms between objects (groups, rings, modules, objects of abelian categories, etc.) such that the image of a morphism equals the kernel of the next.
And a sequence is short exact if in- $$ 0 \rightarrow A\xrightarrow{f} B \xrightarrow{g} C \rightarrow 0$$ our $f$ is injective and $g$ is surjective and $im (f)= ker (g)$.
Actually, short exact of complexes leads to a long exact sequence in homology (let's delve into them at a later time).
Epilogue~
Mathematics often (but not always) begins with the concrete—numbers, shapes, groups—and then dissolves into patterns of relationships (our beloved categories).
The extension problem is a quiet metaphor for this: knowing the parts does not guarantee knowing the whole. We see $N$ and $Q$, yet $G$ resists complete revelation, for structure is more than just the sum. This is the quiet truth mathematics shares with art: form matters as much as substance.
Maybe structure is not an afterthought; but simply the essence.
It is indeed the pattern that breathes life into pieces, and perhaps this is why the language of exactness feels profound—it teaches us that understanding lies not in what we have, but in how what we have holds together.