(Pontrjagin Duality) If G is a (discrete) abelian group, its...

(Pontrjagin Duality) If $G$ is a (discrete) abelian group, its Pontrjagin dual is the group $$ G^{*}=\operatorname{Hom}_{\mathrm{Z}}(G, \mathbb{R} / \mathbb{Z}) . $$ (More generally, the Pontrjagin dual of a locally compact abelian topological group $G$ consists of all the continuous homomorphisms from $G$ into the circle group $S^{1} \cong \mathbb{R} / \mathbb{Z}$.) (i) If $G$ is an abelian group and $a \in G$ is nonzero, prove that there is a homomorphism $f: G \rightarrow \mathbb{R} / \mathbb{Z}$ with $f(a) \neq 0$. (ii) Prove that $\mathbb{R} / \mathbb{Z}$ is an injective abelian group. (iii) Prove that if $0 \rightarrow A \rightarrow G \rightarrow B \rightarrow 0$ is an exact sequence of abelian groups, then so is $0 \rightarrow B^{*} \rightarrow G^{*} \rightarrow A^{*} \rightarrow 0$. (iv) If $G$ is a finite abelian group, then $G^{*} \cong \operatorname{Hom}_{\mathrm{Z}}(G, Q / \mathbb{Z})$. (v) If $G$ is a finite abelian group, prove that $G^{*} \cong G$. (vi) Prove that every quotient group $G / H$ of a finite abelian group $G$ is isomorphic to a subgroup of $G$. Remark. The analogous statement for nonabelian groups is false: if $Q$ is the group of quaternions, then $Q / Z(Q) \cong$ $\mathbf{V}$, where $Z(\mathbf{Q})$ is the center of $\mathbf{Q}$ and $\mathbf{V}$ is the four-group. But $\mathbf{Q}$ has only one element of order 2 while $\mathbf{V}$ has three elements of order 2, so that $\mathbf{V}$ is not isomorphic to a subgroup of $\mathbf{Q}$. Part (vi) is also false for infinite abelian groups: since $\mathbb{Z}$ has no element of order 2 , it has no subgroup isomorphic to $\mathbb{Z} / 2 \mathbb{Z}=\mathrm{I}_{2}$.

(Pontrjagin Duality) If $G$ is a (discrete) abelian group, its Pontrjagin dual is the group
$$
G^{*}=\operatorname{Hom}_{\mathrm{Z}}(G, \mathbb{R} / \mathbb{Z}) .
$$
(More generally, the Pontrjagin dual of a locally compact abelian topological group $G$ consists of all the continuous homomorphisms from $G$ into the circle group $S^{1} \cong \mathbb{R} / \mathbb{Z}$.)
(i) If $G$ is an abelian group and $a \in G$ is nonzero, prove that there is a homomorphism $f: G \rightarrow \mathbb{R} / \mathbb{Z}$ with $f(a) \neq 0$.
(ii) Prove that $\mathbb{R} / \mathbb{Z}$ is an injective abelian group.
(iii) Prove that if $0 \rightarrow A \rightarrow G \rightarrow B \rightarrow 0$ is an exact sequence of abelian groups, then so is $0 \rightarrow B^{*} \rightarrow G^{*} \rightarrow A^{*} \rightarrow 0$.
(iv) If $G$ is a finite abelian group, then $G^{*} \cong \operatorname{Hom}_{\mathrm{Z}}(G, Q / \mathbb{Z})$.
(v) If $G$ is a finite abelian group, prove that $G^{*} \cong G$.
(vi) Prove that every quotient group $G / H$ of a finite abelian group $G$ is isomorphic to a subgroup of $G$.
Remark. The analogous statement for nonabelian groups is false: if $Q$ is the group of quaternions, then $Q / Z(Q) \cong$ $\mathbf{V}$, where $Z(\mathbf{Q})$ is the center of $\mathbf{Q}$ and $\mathbf{V}$ is the four-group. But $\mathbf{Q}$ has only one element of order 2 while $\mathbf{V}$ has three elements of order 2, so that $\mathbf{V}$ is not isomorphic to a subgroup of $\mathbf{Q}$. Part (vi) is also false for infinite abelian groups: since $\mathbb{Z}$ has no element of order 2 , it has no subgroup isomorphic to $\mathbb{Z} / 2 \mathbb{Z}=\mathrm{I}_{2}$.

Key Concepts

Duality for Finite Abelian Groups

For finite abelian groups the Pontrjagin duality reveals that the dual group is isomorphic to the original group. This self-duality is a powerful result that connects the algebraic structure of finite abelian groups with their group of characters. It has implications in various areas of mathematics, such as number theory and coding theory, and provides a clear illustration of how the duality theory simplifies in the finite setting.

Exact Sequences in Abelian Groups

Exact sequences are sequences of group homomorphisms where the image of one homomorphism equals the kernel of the next. In the context of Pontrjagin duality, exact sequences help in understanding how duality functor translates short exact sequences of groups into exact sequences of their duals. This tool is crucial in the study of homological aspects of duality and in proving results about the correspondence between subgroups and quotient groups.

Injective Abelian Groups

Injective abelian groups are those which have the property that any homomorphism defined on a subgroup of an abelian group can be extended to the whole group. This concept is significant in homological algebra and plays a role in duality theories. In Pontrjagin duality, recognizing that the target group (such as ?/?) is injective has important consequences for the ability to extend characters and to analyze exact sequences of groups.

Locally Compact Abelian Groups

Locally compact abelian groups serve as the underlying structures for Pontrjagin duality. These groups, which may be discrete, compact, or otherwise, possess a topology that allows for the application of analysis methods. Their local compactness ensures the existence of a Haar measure, and their abelian (commutative) nature simplifies many aspects of representation theory, as the group structure interacts well with functional analytic techniques.

Pontrjagin Duality

Pontrjagin duality is a fundamental concept in harmonic analysis and topological group theory which associates to a locally compact abelian group G its dual group G*, defined as the group of all continuous homomorphisms (called characters) from G into the circle group S¹. This duality establishes a bridge between the structure of the group and the properties of functions defined on it, often leading to reflexivity results whereby a group can be recovered from its double dual.

Group Homomorphisms into the Circle Group

A central idea in the study of Pontrjagin duality is the use of homomorphisms from a given abelian group to the circle group S¹ (or equivalently ?/?). These homomorphisms, known as characters, translate abstract group elements into complex exponentials. Such maps are crucial not only for the duality theory, but also for understanding properties like the separation of points, as every non-zero element in an abelian group can often be 'detected' by some character.

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