Quick definition question: Dihedral group

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SUMMARY

The dihedral group of an n-gon, denoted as Dn, has an order of 2n, representing the symmetries of the polygon. For example, the dihedral group D4 corresponds to a square with 8 symmetries, while D8 for an octagon has 16 symmetries. To construct the multiplication table for D4, utilize the relations S² = 1, R⁴ = 1, and SR = R³S, which simplifies the process of finding the table. This method effectively organizes the reflections and rotations associated with the square's symmetries.

PREREQUISITES
  • Understanding of group theory concepts, specifically dihedral groups.
  • Familiarity with symmetry operations in geometry.
  • Knowledge of mathematical notation for group relations.
  • Basic skills in constructing multiplication tables for algebraic structures.
NEXT STEPS
  • Study the properties of dihedral groups in detail, focusing on Dn for various n.
  • Learn how to derive multiplication tables for other dihedral groups, such as D6 and D8.
  • Explore applications of dihedral groups in symmetry and geometry.
  • Investigate the relationship between dihedral groups and other algebraic structures, such as cyclic groups.
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Mathematicians, students of abstract algebra, and anyone interested in the study of symmetries in geometry will benefit from this discussion.

srfriggen
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A dihedral group of an n-gon denoted by Dn, whose corresponding group is called the Dihedral group of order 2n?


What I gather from that is a square has 8 symmetries, an octagon has 16, a hexagon 12, etc?
 
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Yep, you got it.
 
When I do reflection in square do I need to put numbers and that do reflection for particular site of square. Is there some easiest way to find multiplication table of ##D_4##?
 
LagrangeEuler said:
When I do reflection in square do I need to put numbers and that do reflection for particular site of square. Is there some easiest way to find multiplication table of ##D_4##?

Let ##R## be a rotation and ##S## a reflection. Use the relations ##S^2 = 1##, ##R^4 = 1## and ##SR = R^3S##. Using this you can find the multiplication table easily.
 

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