(i) If A is an abelian group with m A=A for some nonzero m ∈ℤ, prove that every exact sequence 0

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(i) If A is an abelian group with m A=A for some nonzero m...

(i) If $A$ is an abelian group with $m A=A$ for some nonzero $m \in \mathbb{Z}$, prove that every exact sequence $0 \rightarrow A \rightarrow G \rightarrow$ $\mathbb{I}_{m} \rightarrow 0$ splits. Conclude that $m \operatorname{Ext}_{\mathbb{Z}}^{1}(A, B)=\{0\}=$ $m \operatorname{Ext}_{\mathbb{Z}}^{1}(B, A)$. (ii) If $A$ and $C$ are abelian groups with $m A=\{0\}=n C$, where $(m, n)=1$, prove that every extension of $A$ by $C$ splits.

(i) If $A$ is an abelian group with $m A=A$ for some nonzero $m \in \mathbb{Z}$, prove that every exact sequence $0 \rightarrow A \rightarrow G \rightarrow$ $\mathbb{I}_{m} \rightarrow 0$ splits. Conclude that $m \operatorname{Ext}_{\mathbb{Z}}^{1}(A, B)=\{0\}=$ $m \operatorname{Ext}_{\mathbb{Z}}^{1}(B, A)$.
(ii) If $A$ and $C$ are abelian groups with $m A=\{0\}=n C$, where $(m, n)=1$, prove that every extension of $A$ by $C$ splits.

Key Concepts

Ext Functor in Homological Algebra

The Ext functor, and specifically Ext¹, measures the equivalence classes of extensions of one abelian group by another. It encapsulates the obstructions to splitting a short exact sequence. When properties such as divisibility ensure that multiplication by m on Ext¹ results in the zero element, it demonstrates that the possible non-trivial extensions are annihilated by m, linking the algebraic structure of the groups with the behavior of their extensions.

Divisibility in Abelian Groups

A group is said to be divisible by an integer m if every element of the group can be expressed as m times some other element in the group. This property is critical in proving that certain exact sequences split, as it provides the necessary flexibility to construct splitting maps by solving equations of the form m·x = a within the group.

Coprime Integers in Group Extensions

When dealing with two groups where one is annihilated by m and the other by n, and m and n are coprime, the lack of common factors ensures that their interactions in an extension are minimal. This coprimality condition is a key tool in proving that every extension in such a setting splits, since it allows one to construct a splitting morphism using number-theoretic properties such as the Euclidean algorithm.

Exact Sequence

An exact sequence is a sequence of abelian groups and homomorphisms between them in which the image of one homomorphism coincides with the kernel of the next. Short exact sequences, particularly those of the form 0 ? A ? G ? C ? 0, are used to describe how a group G can be seen as an extension of one group by another, laying the groundwork for analyzing their structure and the possibility of decomposing G into simpler components.

Splitting of Exact Sequences

A short exact sequence splits when there exists a homomorphism (called a section) from the quotient group back into the middle group that effectively 'reverses' the surjection, leading to an isomorphism between G and the direct sum (or direct product) of A and C. Splitting is a central concept because it indicates that the extension is trivial, meaning there are no hidden interactions between A and C within G.

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