https://www.dpmms.cam.ac.uk/~wtg10/definition.html

With thanks and regards to Prof. Tim Gowers,

Nalin Pithwa

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# Month: August 2019

# Definition of definition : Prof. Tim Gowers, Fields Medalist

# You and your research (you and your studies) by Richard Hamming, AT & T, Bell Labs mathematician.

# Dihedral groups explained by I N Herstein

https://www.dpmms.cam.ac.uk/~wtg10/definition.html

With thanks and regards to Prof. Tim Gowers,

Nalin Pithwa

Reference: Abstract Algebra, Third Edition, I N Herstein

First consider the following: Let S be the plane, that is, and consider defined by and ; f is the reflection about the y-axis and g is the rotation through 90 degrees in a counterclockwise direction about the origin. We then define , and let * in G be the product of elements in A(S). Clearly, identity mapping;

and

So .

It is a good exercise to verify that and G is a non-abelian group of order 8. This group is called the dihedral group of order 8. [Try to find a formula for that expresses a, b in terms of i, j, s and t.

II) Let S be as in above example and f the mapping in above example. Let and let h be the rotation of the plane about the origin through an angle of in the counterclockwise direction. We then define and define the product * in G via the usual product of mappings. One can verify that identity mapping and . These relations allow us to show that (with some effort) G is a non-abelian group of order 2n. G is called the dihedral group of order 2n.

More later,

Nalin Pithwa

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