Direct product of two groups with different n-spaces

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SUMMARY

The discussion focuses on evaluating the direct product of two groups, specifically a group G that is a direct product of two groups A and B, and a group H. The conclusion is that the direct product G x H can be expressed as A x B x H, demonstrating that the structure remains consistent regardless of the dimensionality of the components. The notation (a, b, h) is used to represent elements from groups A, B, and H, respectively, illustrating the relationship between the groups. This is analogous to the mathematical representation of R^3 as R x R^2.

PREREQUISITES
  • Understanding of group theory and direct products
  • Familiarity with notation for tuples in mathematical contexts
  • Knowledge of the properties of Cartesian products
  • Basic comprehension of vector spaces and their dimensions
NEXT STEPS
  • Study the properties of direct products in group theory
  • Learn about the implications of group isomorphisms
  • Explore the relationship between groups and vector spaces
  • Investigate examples of direct products in algebraic structures
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Mathematicians, students of abstract algebra, and anyone interested in the structural properties of groups and their products.

huey910
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how does one evaluate the direct product between a group G with components that are say 2-tuple and a group H with components that are just 1-tuple?
 
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I'm sorry, I have no single idea what you mean. Could you please clarify?? Maybe use some notation?
 
I guess he means taking the direct product of a group G and a group H, where G is already known to be a direct product of two groups A and B , and H is just another group: i.e. G x H. In this case, the direct product isn't anything more special, as G x H = A x B x H since ( A x B ) x H = A x ( B x H ) .. et c. Then, you can write ( a, b, h ) where a is in A , b is in B and h is in H. Just think about R^3 = R x R^2
 

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