user: Extend Algebraic Structures Week 1 with a short section on the last ten slides of the slideshow pdf linked at the end of the note, where the slides talk about Direct Products of groups/sets and if they’re groups, with examples
ai: Direct Products of Groups/Sets
The direct product of two algebraic structures, such as groups or sets, is a way to combine them into a new structure. If we have two groups ( G ) and ( H ), their direct product, denoted ( G \times H ), consists of all ordered pairs ( (g, h) ) where ( g \in G ) and ( h \in H ). The operation on this product is defined component-wise.
Definition
For ( G ) and ( H ) to form a direct product ( G \times H ), the following must hold:
- Closure: If ( (g_1, h_1), (g_2, h_2) \in G \times H ), then ( (g_1, h_1) * (g_2, h_2) = (g_1 * g_2, h_1 * h_2) ) is also in ( G \times H ).
- Associativity: The operation is associative, meaning ( ((g_1, h_1) * (g_2, h_2)) * (g_3, h_3) = (g_1, h_1) * ((g_2, h_2) * (g_3, h_3)) ).
- Identity Element: The identity element of ( G \times H ) is ( (e_G, e_H) ), where ( e_G ) and ( e_H ) are the identity elements of ( G ) and ( H ), respectively.
- Inverses: For each ( (g, h) \in G \times H ), there exists an inverse ( (g^{-1}, h^{-1}) ) such that ( (g, h) * (g^{-1}, h^{-1}) = (e_G, e_H) ).
Examples
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Example with Groups: Let ( G = \mathbb{Z}/2\mathbb{Z} ) (the integers modulo 2) and ( H = \mathbb{Z}/3\mathbb{Z} ) (the integers modulo 3). The direct product ( G \times H ) consists of the pairs ( (0,0), (0,1), (0,2), (1,0), (1,1), (1,2) ). This set forms a group under component-wise addition modulo 2 and 3, respectively.
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Example with Sets: Consider two sets ( A = {1, 2} ) and ( B = {x, y} ). The direct product ( A \times B ) is the set of ordered pairs ( {(1, x), (1, y), (2, x), (2, y)} ). While this is not a group, it illustrates the concept of combining sets.
Conclusion
The direct product of groups is itself a group, provided that the original structures are groups. This construction allows for the exploration of more complex algebraic structures by combining simpler ones, enriching our understanding of their properties and behaviors.