If A and B are (not necessarily abelian) groups, prove that A ∩B= A ×B (direct product) in Group

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If A and B are (not necessarily abelian) groups, prove that...

If $A$ and $B$ are (not necessarily abelian) groups, prove that $A \cap B=$ $A \times B$ (direct product) in Groups. For readers familiar with group theory, prove that $A \sqcup B=A * B$ (free product) in Groups.

Key Concepts

Direct Product as Categorical Product

The direct product of groups, denoted by A × B, is defined as the set of all ordered pairs (a, b) with a in A and b in B, equipped with the componentwise operation. Within category theory, this construction is characterized as the categorical product because it satisfies the universal property: for any group X along with homomorphisms f: X ? A and g: X ? B, there exists a unique homomorphism h: X ? A × B such that the projections composed with h yield f and g respectively. This notion of a product, defined purely through its mapping property, is important in abstract algebra as it unifies various product-like constructions across different categories.

Free Product as Categorical Coproduct

The free product of groups, denoted by A * B, is the coproduct in the category of groups. Unlike the direct product, the free product imposes no relations between the subgroups A and B beyond those inherent within each group. Its universal property states that for any pair of homomorphisms from A and B to another group G, there exists a unique homomorphism from A * B to G that extends these maps. The free product captures the idea of amalgamating groups in the freest possible way that maintains the structure of each, and serves as a fundamental tool when studying constructions that involve ‘gluing’ together groups.

Universal Property

Central to both the direct product and free product constructions is the concept of the universal property, which provides an abstract, categorical definition of these objects. A universal property identifies an object (up to isomorphism) in terms of its unique mapping characteristics with respect to other objects and morphisms in the category. For instance, the universal property for the direct product guarantees that any pair of homomorphisms from a given group to A and B factors uniquely through A × B, while the universal property for the free product ensures that any pair of homomorphisms from A and B into another group uniquely extends to a homomorphism from A * B.

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