Question on group theory: simplest math construction

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SUMMARY

The discussion focuses on the construction of the dihedral group D_2 and its subgroups, specifically addressing why only two of the three subgroups Z_2 are utilized in certain mathematical constructions. The participants analyze the implications of using all three subgroups, concluding that a direct product group formed from three subgroups, each containing two elements, would yield eight distinct elements. This outcome indicates that such a construction cannot be isomorphic to a group with only four elements, which aligns with the author's goal of representing a group with four elements.

PREREQUISITES
  • Understanding of group theory concepts, particularly dihedral groups.
  • Familiarity with subgroup structures and their properties.
  • Knowledge of isomorphism in group theory.
  • Basic comprehension of direct product groups and their element counts.
NEXT STEPS
  • Study the properties of dihedral groups, specifically D_2 and its subgroups.
  • Learn about the concept of group isomorphism and its implications in group theory.
  • Explore the construction and characteristics of direct product groups in detail.
  • Investigate examples of subgroup structures within various groups to solidify understanding.
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Mathematicians, students of abstract algebra, and anyone interested in the foundational concepts of group theory and its applications in mathematical constructions.

Abolaban
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Hello Big Minds,

In the following analysis...It is said that D_2 contains three subgroups Z_2...why did he choose a mathematical constuction contains only two of the the three subgroups? shouldn't he use the three in his construction? what will happen if he used the three?

upload_2015-2-1_10-52-31.png

[from the book of P.Ramond, Group Theory, p8.]

best regards

Abolaban
 
Last edited:
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How many distinct elements would the direct product group have if he used 3 subgroups, each with 2 elements in them? If a direct product has 8 elements , it can't be isomorphic to a group with only 4 elements in it. His goal was to write an direct product that is essentially the same group as the 4 element group.
 

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