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Construction junctions are diamonds. They show where multiple structures must coexist on the same carrier and satisfy compatibility conditions.

Canonical static concept record

Abelian group

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Summary

A group with commutative multiplication, usually written additively.

Record metadata

Canonical name: Abelian group
Concept ID: abelian_group
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Algebra
Domains: Algebra
Alternate terminology: No alternate terminology is currently authored for this record.

Carrier(s)

set $A$

Data

group operation

Axioms / constraints

group axioms
commutativity

Concept sources

Wikipedia — Abelian group — Abelian group · expository · source ID wp-abelian
nLab — Abelian group — Abelian group · mathematical reference · source ID nlab-abelian-group
Stacks Project — Commutative Algebra (PDF) — Commutative Algebra (PDF) · reference text · source ID stacks-algebra-text

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Cohomology group Abelian group

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: story_cohomology_group_to_abelian_group_induced_forgotten
Relation type: Induced / forgotten induced-forgotten
Direction: source → target
Endpoint roles: source: Yields by induction / forgetting; target: Obtained by induction / forgetting from
Authored annotation: is an abelian group

Authored explanation

Cohomology groups are abelian groups under addition induced from the cochain groups.

How to interpret this relation type

Pass canonically from a stronger object to structure it determines, or forget part of the data while retaining a valid weaker structure. The carrier may change under a canonical induced construction.

Relation sources

Hatcher — Algebraic Topology — Algebraic Topology · textbook · source ID hatcher-algebraic-topology
Wikipedia — Abelian group — Abelian group · expository · source ID wp-abelian
Stacks Project — Commutative Algebra (PDF) — Commutative Algebra (PDF) · reference text · source ID stacks-algebra-text

Group Abelian group

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: e_group_abelian
Relation type: Impose axiom impose-axiom
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: + commutativity

Authored explanation

Require $xy=yx$ .

How to interpret this relation type

Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.

Relation sources

Wikipedia — Abelian group — Abelian group · expository · source ID wp-abelian
Stacks Project — Commutative Algebra (PDF) — Commutative Algebra (PDF) · reference text · source ID stacks-algebra-text

Higher homotopy group $\pi_n(X,x_0)$ Abelian group

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: story_higher_homotopy_group_to_abelian_group_induced_forgotten
Relation type: Induced / forgotten induced-forgotten
Direction: source → target
Endpoint roles: source: Yields by induction / forgetting; target: Obtained by induction / forgetting from
Authored annotation: is abelian for $n\ge2$

Authored explanation

The homotopy group $\pi_n$ is abelian for every $n\ge2$ .

How to interpret this relation type

Pass canonically from a stronger object to structure it determines, or forget part of the data while retaining a valid weaker structure. The carrier may change under a canonical induced construction.

Relation sources

Hatcher — Algebraic Topology — Algebraic Topology · textbook · source ID hatcher-algebraic-topology
Wikipedia — Abelian group — Abelian group · expository · source ID wp-abelian
Stacks Project — Commutative Algebra (PDF) — Commutative Algebra (PDF) · reference text · source ID stacks-algebra-text

Homology group Abelian group

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: story_homology_group_to_abelian_group_induced_forgotten
Relation type: Induced / forgotten induced-forgotten
Direction: source → target
Endpoint roles: source: Yields by induction / forgetting; target: Obtained by induction / forgetting from
Authored annotation: is an abelian group

Authored explanation

Homology groups are abelian groups under addition induced from the chain groups.

How to interpret this relation type

Pass canonically from a stronger object to structure it determines, or forget part of the data while retaining a valid weaker structure. The carrier may change under a canonical induced construction.

Relation sources

Hatcher — Algebraic Topology — Algebraic Topology · textbook · source ID hatcher-algebraic-topology
Wikipedia — Abelian group — Abelian group · expository · source ID wp-abelian
Stacks Project — Commutative Algebra (PDF) — Commutative Algebra (PDF) · reference text · source ID stacks-algebra-text

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.

Abelian group Commutative monoid

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: abelian_group_to_commutative_monoid
Relation type: Induced / forgotten induced-forgotten
Direction: source → target
Endpoint roles: source: Yields by induction / forgetting; target: Obtained by induction / forgetting from
Authored annotation: forget additive inverses

Authored explanation

Retain the commutative associative operation and identity.

How to interpret this relation type

Pass canonically from a stronger object to structure it determines, or forget part of the data while retaining a valid weaker structure. The carrier may change under a canonical induced construction.

Relation sources

Wikipedia — Abelian group — Abelian group · expository · source ID wp-abelian
Wikipedia — Commutative monoid — Commutative monoid · expository · source ID wp-comm-monoid
Stacks Project — Commutative Algebra (PDF) — Commutative Algebra (PDF) · reference text · source ID stacks-algebra-text

Abelian group ring action on an abelian group(construction junction)

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: e_ab_jmodule
Relation type: Combine compatible structures combine-compatible
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: additive carrier $M$

Authored explanation

Supply the additive abelian group.

How to interpret this relation type

Feed several structures into a construction junction and impose compatibility between them.

Relation sources

Wikipedia — Module (mathematics) — Module (mathematics) · expository · source ID wp-module
Stacks Project — Commutative Algebra (PDF) — Commutative Algebra (PDF) · reference text · source ID stacks-algebra-text

Abelian group field action on an abelian group(construction junction)

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: e_ab_jvector
Relation type: Combine compatible structures combine-compatible
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: additive carrier $V$

Authored explanation

Supply the additive abelian group.

How to interpret this relation type

Feed several structures into a construction junction and impose compatibility between them.

Relation sources

Wikipedia — Vector space — Vector space · expository · source ID wp-vector
Stacks Project — Commutative Algebra (PDF) — Commutative Algebra (PDF) · reference text · source ID stacks-algebra-text