Need Help in Direct Product and Quotient (factor) Group

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SUMMARY

The discussion centers on abstract algebra, specifically the properties of direct products and quotient groups. It confirms that the direct product of Z3 and Z7 equals Z21, while Z4 and Z2 combine to form Z8, and Z2 x Z2 equals Z4. Additionally, it clarifies that Z2 x Z4 represents a Cartesian product, and Z/nZ is indeed a quotient group, exemplified by Z/5Z equating to Z5, which consists of the elements {0, 1, 2, 3, 4} under addition modulo 5.

PREREQUISITES
  • Understanding of group theory concepts such as direct products and quotient groups.
  • Familiarity with modular arithmetic and notation like Z/nZ.
  • Knowledge of Cartesian products in set theory.
  • Basic proficiency in abstract algebra terminology and theorems.
NEXT STEPS
  • Study the properties of direct products in group theory.
  • Learn about quotient groups and their applications in abstract algebra.
  • Explore modular arithmetic in greater depth, particularly Z/nZ structures.
  • Investigate Cartesian products and their implications in algebraic structures.
USEFUL FOR

Students and educators in mathematics, particularly those focusing on abstract algebra, as well as researchers looking to deepen their understanding of group theory concepts.

Kenji Liew
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Hi All. I have several questions on abstract algebra.

Here are my questions and the attempts I had done so far.

(1)Let denote Z as the integer.According to theorem, the direct product of Z3 X Z7 = Z21.
Hence, is Z4 X Z2 is equal to Z8?
Z2X Z2 is equal to Z4?

(2)For Z4 ={0,1,2,3 }, and Z2 ={0,1}.
Then what are the elements will be for Z2 x Z4? Is this a Cartesian product?

(3) We know Z/nZ =Zn, it's a quotient/factor group where nz can be defined as (aH)(bH) = (ab)H. To have a numerical example, we have Z/5Z= Z5.
As usual Z5= {0,1,2,3,4}. But what is the elements contain in Z/5Z and how to find the elements so that the Z/5Z= Z5?

I need some explanations to make me clearer. Thanks a lot ;)
 
Last edited:
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(3) Just consider addition in Z modulo 5.
 

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