Commutator subgroup; quaternions; cyclic groups


image

from http://mathworld.wolfram.com/CommutatorSubgroup.html

image

from http://mathworld.wolfram.com/Commutator.html

Interestingly,

image

 

 

 

 

 

 

 

 

 

 

 

From http://en.wikipedia.org/wiki/Alternating_group#Examples_S4_and_A4

Concerning groups we have been working with, its just interesting that:

image

http://en.wikipedia.org/wiki/Commutator_subgroup

Concerning 1940s ciphering math

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http://unapologetic.wordpress.com/2007/03/14/commutator-subgroups/

An interesting looking book dedicated to quaternions is

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http://read.pudn.com/downloads147/ebook/636697/Visualizing%20Quaternions.pdf

And, finally its interesting to see what happens when one moves the coordinates around.

image image

obviously, we can now do our rotation thing again:

image

And, then we get to the subgroup of 3 elements:

imageimage

Generalizing,

image

From image

concerning quaternion “applicability” in ciphering:

image

From image

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