(i) Prove that a short exact sequence in R Mod, 0 →A i→ B p→ C →0 splits if and only if there e

(i) Prove that a short exact sequence in R Mod, 0 →A i→ B...

(i) Prove that a short exact sequence in ${ }_{R}$ Mod, $$ 0 \rightarrow A \stackrel{i}{\rightarrow} B \stackrel{p}{\rightarrow} C \rightarrow 0 $$ splits if and only if there exists $q: B \rightarrow A$ with $q i=1_{A}$. (Note that $q$ is a retraction $B \rightarrow$ im $i$.) (ii) A sequence $A \stackrel{i}{\rightarrow} B \stackrel{p}{\rightarrow} C$ in Groups is exact if im $i=$ ker $p$; an exact sequence $$ 1 \rightarrow A \stackrel{i}{\rightarrow} B \stackrel{p}{\rightarrow} C \rightarrow 1 $$ in Groups is split if there is a homomorphism $j: C \rightarrow B$ with $p j=1_{C}$. Prove that $1 \rightarrow A_{3} \rightarrow S_{3} \rightarrow \mathbb{I}_{2} \rightarrow 1$ is a split exact sequence. In contrast to part (i), show, in a split exact sequence in Groups, that there may not be a homomorphism $q: B \rightarrow A$ with $q i=1_{A}$.

(i) Prove that a short exact sequence in ${ }_{R}$ Mod,
$$
0 \rightarrow A \stackrel{i}{\rightarrow} B \stackrel{p}{\rightarrow} C \rightarrow 0
$$
splits if and only if there exists $q: B \rightarrow A$ with $q i=1_{A}$.
(Note that $q$ is a retraction $B \rightarrow$ im $i$.)
(ii) A sequence $A \stackrel{i}{\rightarrow} B \stackrel{p}{\rightarrow} C$ in Groups is exact if im $i=$ ker $p$; an exact sequence
$$
1 \rightarrow A \stackrel{i}{\rightarrow} B \stackrel{p}{\rightarrow} C \rightarrow 1
$$
in Groups is split if there is a homomorphism $j: C \rightarrow B$ with $p j=1_{C}$. Prove that $1 \rightarrow A_{3} \rightarrow S_{3} \rightarrow \mathbb{I}_{2} \rightarrow 1$ is a split exact sequence. In contrast to part (i), show, in a split exact sequence in Groups, that there may not be a homomorphism $q: B \rightarrow A$ with $q i=1_{A}$.

Key Concepts

Short Exact Sequence

A short exact sequence is a chain of homomorphisms between objects (such as modules or groups) of the form 0 ? A ? B ? C ? 0, where the image of each map equals the kernel of the next. This concept is fundamental in homological algebra as it captures the idea of how a structure (B) is built from a subobject (A) and a quotient (C), and sets the stage for discussing extensions, splittings, and the classification of objects in an abelian category.

Split Exact Sequence

A split exact sequence is one in which the exact sequence can be broken up into simpler pieces, meaning that the middle object is isomorphic to the direct sum (or semidirect product in nonabelian settings) of the subobject and the quotient. In the category of modules, a short exact sequence splits if there exists a retraction or a section, which implies that the object B can be decomposed as A ? C. In more general categories such as groups, a split sequence involves similar conditions but may have additional subtleties, as the existence of a splitting homomorphism does not always provide a retraction in the same way.

Retraction and Section Maps

Retractions and sections are map-theoretic tools used to undo or split homomorphisms. A retraction is a map q: B ? A such that q composed with the inclusion map from A to B is the identity on A, while a section is a map going in the opposite direction that serves as a right inverse to the projection map. Their existence is equivalent to the splitting of a short exact sequence in many contexts (for example, in module categories), allowing one to express the middle object as a combination of the kernel and the image.

Semidirect Product

The semidirect product is a construction that generalizes the direct sum or direct product in the context of nonabelian groups. It arises naturally when a group extension splits, i.e., when the group can be seen as being built from a normal subgroup and another subgroup that 'acts' on it. While module extensions that split yield a direct sum, group extensions that split yield a semidirect product, highlighting how the internal structure and the action of the quotient on the kernel influence the overall group structure.

Transcript

00:01 Get the general linear group of n -by -n -n -n -singular matrices under matrix multiplication, specifically with matrix elements in the real numbers.

00:11 And we're considering r -star, that's the group of non -zero real numbers under scalar multiplication.

00:21 We're going to define this map from gl to r -star, such that, so fee of some matrix, just returns the determinant of the matrix.

00:32 We want to show that fee is a homomorphism and that the kernel of fee is a subgroup of gl.

00:39 And also then we want to look at the cosets of the subgroup k and show that the cosets form a group.

00:49 So let's first show that this is a homomorphism.

00:55 We can first show that this map is going to hold, that this is in fact a map.

01:00 Between the two.

01:02 Remember then since these are all non -singular matrices for all m elements of gl and r, we know that the determinant of m is not going to be equal to zero and will be an element of non -zero numbers since the elements are real as well.

01:26 So real numbers times real numbers, and they can't combine to equal zero.

01:32 So that's good.

01:35 Next, let's look at, let's show that the identity maps to the identity.

01:42 So the identity and gl is just the identity matrix.

01:45 If we consider fee of that, we're looking at the determinant of i.

01:50 Let's write it like this.

01:52 That is equal to one, which is the identity element.

01:58 In our star.

02:00 So that works out.

02:01 Now let's look at multiplication...