Symmetric colorings of finite groups
Date
2018
Authors
Phakathi, Jabulani
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Abstract
Given a finite group G and r ∈ N, an r-coloring(or coloring) of G is a mapping χ : G −→ {1,2,3,...,r}. The group G naturally acts on its colorings by χ(xg−1) = χ(x). Colorings χ and ψ are equivalent if there is g ∈ G such that χ(xg−1) = ψ(x) for all x ∈ G. A coloring χ of G is called symmetric if there is g ∈ G such that χ(gx−1g) = χ(x) for all x ∈ G. Let |Sr(G)| denote the number of symmetric rcolorings of G and |Sr(G)/ ∼ | the number of equivalence classes of symmetric r-colorings of G. We present methods for computing |Sr(G)/ ∼ | and |Sr(G)| and derive explicit formulas in some cases, in particular cyclic group Zn and the dihedral group Dn.
Description
A Thesis presented for the degree of Doctor of Philosophy, 2018
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Citation
Phakathi, Jabulani (2018) Symmetric colorings of finite groups, University of the Witwatersrand, Johannesburg, <http://hdl.handle.net/10539/26656>