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

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

Interestingly,

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

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

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

Concerning 1940s ciphering math

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

An interesting looking book dedicated to quaternions is

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

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

obviously, we can now do our rotation thing again:

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

Generalizing,

From

concerning quaternion “applicability” in ciphering:

From

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