Constructing Isomorphisms for D4 and A4

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SUMMARY

The discussion focuses on constructing an isomorphism between the group A4 (the alternating group on four elements) and D4 (the dihedral group of symmetries of a square), resulting in the structure (Z/2,+) x (Z/2,+). The isomorphism is established by mapping the identity element of D4 to (0,0) in (Z/2,+) x (Z/2,+) and mapping the elements (13)(24), (12)(34), and (14)(23) to (0,1), (1,0), and (1,1) respectively. This construction confirms the isomorphic relationship between these groups.

PREREQUISITES
  • Understanding of group theory concepts, specifically A4 and D4.
  • Familiarity with isomorphisms in abstract algebra.
  • Knowledge of the structure of (Z/2,+) x (Z/2,+).
  • Basic understanding of permutation notation and operations.
NEXT STEPS
  • Study the properties of A4 and D4 in detail.
  • Learn about group isomorphisms and their applications in abstract algebra.
  • Explore the implications of the structure (Z/2,+) x (Z/2,+) in group theory.
  • Investigate other examples of isomorphic groups and their constructions.
USEFUL FOR

Mathematicians, students of abstract algebra, and anyone interested in group theory and its applications in symmetry and structure analysis.

PhysicsHelp12
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Construct an isomorphism

A4 (AND) D4 ---> (Z/2,+)X(Z/2,+)

where D4's the symmetries of the square

so I found D4(And)A4={(13)(24),Id,(12)(34),(14)(23)}
 
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Then you are trying too hard! Map the identity to the identity of (Z/2,+)x(Z/2,+), (0,0). Then map (13)(24), (12)(34), (14)(23) to (0,1), (1,0), (1,1) any way you like. You will get an isomorphism.
 

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