What is the Composition Calculation Method for External Direct Products?

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Discussion Overview

The discussion revolves around the composition calculation method for external direct products in group theory. Participants explore the symbolism and structure of the operation, as well as the specific example involving the groups U(8) and U(10).

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant expresses confusion about the symbolism and structure of external direct products, questioning the example provided in the text.
  • Another participant clarifies that GxH consists of ordered pairs (g,h) and describes the operation for composing elements of G and H.
  • A participant reiterates their confusion, suggesting that the example's structure implies a distribution of elements that does not align with the definition of ordered pairs.
  • Another participant challenges this interpretation, emphasizing the importance of ordered pairs in the definition of the direct product and pointing out the differences in the formats discussed.

Areas of Agreement / Disagreement

Participants do not reach a consensus, as there is ongoing confusion and differing interpretations of the structure and calculation method for external direct products.

Contextual Notes

There are unresolved aspects regarding the interpretation of the example and the notation used, which may depend on the definitions and conventions adopted by different sources.

williamaholm
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greetings, I'm confused on how to perform external direct products. I'm not sure how to symbolize this operation so I'll use +.
My text explains it as: G1+G2+...+Gn ={(g1,g2,...gn)}|giEGi}, where (g1,g2,...,gn)(g'1,g'2,...,g'n) is defined as (g1g'1, g2g'2,..., gng'n). It then gives this as an example: U(8)+U(10) = {(1,1),(1,3),(1,7),(1,9),(3,1),(3,3),(3,7),(3,9),(5,1),(5,3),(5,7),(5,9),(7,1),(7,3),(7,7),(7,9)} . Since U(8)={1,3,5,7} and U(10)={1,3,7,9} doesn't this example have the form of (g1g'1, g1g'2, g1g'3, g1g'4, g2g'1, g2g'2, g2g'3... and so forth until we cycled through it all) :bugeye:
 
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GxH is the set of all ordered pairs (g,h) where g is in G and h in H. Which is what you've written. And?
It is a group under the operation (g,h)*(g',h')=(g*g',h*h') where we compose elements of G in the obvious way, and elements of H similarly.

I can't decipher what you think is wrong.
 
still confused

"doesn't this example have the form of (g1g'1, g1g'2, g1g'3, g1g'4, g2g'1, g2g'2, g2g'3..."
g1 is being distributed among all of g' s elements and then g2 gets distributed among all of g' s elements and this continues for all g elements. rather than g1g'1, g2g'2, g3g'3, where each element in g is associated with only one element in g'. Does this help you see where I'm confused?
 
No. Clearly the list you give with numbers in has more commas and different brackets. The list with g1 etc is a completely different format.

g1 distributed amongst the g's? what g's?

Read the definition of what the elements in the direct product are (ORDERED PAIRS, you have no ordered pairs mentioned in post #3.)


this bit here:
(g1g'1, g2g'2,..., gng'n)

refers to how you calculate the composition of two elements in the direct product.
 

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