Interesting Esoterica

Amusing Permutation Representations of Group Extensions

Article by Yongju Bae and J. Scott Carter and Byeorhi Kim
  • Published in 2018
  • Added on
Wreath products of finite groups have permutation representations that are constructed from the permutation representations of their constituents. One can envision these in a metaphoric sense in which a rope is made from a bundle of threads. In this way, subgroups and quotients are easily visualized. The general idea is applied to the finite subgroups of the special unitary group of $(2\times 2)$-matrices. Amusing diagrams are developed that describe the unit quaternions, the binary tetrahedral, octahedral, and icosahedral group as well as the dicyclic groups. In all cases, the quotients as subgroups of the permutation group are readily apparent. These permutation representations lead to injective homomorphisms into wreath products.

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BibTeX entry

@article{AmusingPermutationRepresentationsofGroupExtensions,
	title = {Amusing Permutation Representations of Group Extensions},
	abstract = {Wreath products of finite groups have permutation representations that are
constructed from the permutation representations of their constituents. One can
envision these in a metaphoric sense in which a rope is made from a bundle of
threads. In this way, subgroups and quotients are easily visualized. The
general idea is applied to the finite subgroups of the special unitary group of
{\$}(2\times 2){\$}-matrices. Amusing diagrams are developed that describe the unit
quaternions, the binary tetrahedral, octahedral, and icosahedral group as well
as the dicyclic groups. In all cases, the quotients as subgroups of the
permutation group are readily apparent. These permutation representations lead
to injective homomorphisms into wreath products.},
	url = {http://arxiv.org/abs/1812.08475v1 http://arxiv.org/pdf/1812.08475v1},
	year = 2018,
	author = {Yongju Bae and J. Scott Carter and Byeorhi Kim},
	comment = {},
	urldate = {2019-01-01},
	archivePrefix = {arXiv},
	eprint = {1812.08475},
	primaryClass = {math.GT},
	collections = {attention-grabbing-titles,fun-maths-facts}
}