Recall that every abelian group G having no elements of...

Recall that every abelian group $G$ having no elements of infinite order has a primary decomposition: $G=\bigoplus_{p} G_{p}$, where $p$ is a prime and $G_{p}=\{g \in G:$ order $g$ is some power of $p\}$. In particular, the $p$-primary component of $G=\mathbb{Q} / \mathbb{Z}$ is called the Prüfer group; it is denoted by $\mathbb{Z}\left(p^{\infty}\right)$. (i) Prove that $\mathbb{Z}\left(p^{\infty}\right)$ is an injective abelian group. (ii) Prove that the injective envelope $\operatorname{Env}\left(\mathbb{I}_{p^{n}}\right)$ is $\mathbb{Z}\left(p^{\infty}\right)$.

Recall that every abelian group $G$ having no elements of infinite order has a primary decomposition: $G=\bigoplus_{p} G_{p}$, where $p$ is a prime and $G_{p}=\{g \in G:$ order $g$ is some power of $p\}$. In particular, the $p$-primary component of $G=\mathbb{Q} / \mathbb{Z}$ is called the Prüfer group; it is denoted by $\mathbb{Z}\left(p^{\infty}\right)$.
(i) Prove that $\mathbb{Z}\left(p^{\infty}\right)$ is an injective abelian group.
(ii) Prove that the injective envelope $\operatorname{Env}\left(\mathbb{I}_{p^{n}}\right)$ is $\mathbb{Z}\left(p^{\infty}\right)$.

Key Concepts

Divisible Groups

Divisible groups are abelian groups in which every element is divisible by every positive integer, meaning that for any element g and any positive integer n, there exists an element h such that nh = g. This property implies that every divisible group is an injective module over the integers. In the context of abelian groups, the concept is important because divisible groups serve as building blocks for understanding injectivity and extensions.

Injective Abelian Groups

An injective abelian group is one in which every homomorphism from a subgroup of an abelian group to the injective group can be extended to a homomorphism from the entire group. This property, guaranteed by Baer's criterion for abelian groups, plays a central role in module theory and homological algebra. The classification of injective abelian groups is simplified by the fact that every divisible abelian group is injective, making them key examples of such groups.

Prüfer Groups

The Prüfer group, denoted as Z(p?), is the p-primary component of the quotient group Q/Z and consists of all p-power roots of unity. It is an example of an infinite, divisible p-group and is key for understanding torsion phenomena in abelian groups. Its structure as a divisible and p-torsion group makes it a standard example when studying injective objects in the category of abelian groups.

Primary Decomposition in Abelian Groups

The primary decomposition theorem asserts that an abelian group whose elements all have finite order can be decomposed into a direct sum of its p-primary components, where each component consists solely of elements whose orders are powers of a specific prime p. This decomposition is crucial in analyzing the structure of torsion groups, allowing complex groups to be understood as a sum of simpler, prime-based components.

Injective Envelopes

An injective envelope of an abelian group is the smallest injective group that contains the group as a subgroup, making it a minimal essential extension. The concept of the injective envelope is important in homological algebra as it facilitates the construction of injective resolutions and is used to study and classify the structure of modules. The determination of an injective envelope often involves checking minimality criteria and extension properties.

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