Z/7Z made difficult



here’s a hex board
with seven icons on it;
each icon has seven colors;
the seven permutations of
colors-into-hexes (each icon
has seven hexes) can be
considered as the objects
of a cyclic group.
specifically
{
() = \identity
(1234567) = \psi
(1357246)= \psi^2
(1473625)=\psi^3
(1526374)=\psi^4
(1642753)=\psi^5
(1765432)=\psi^6
}.

(of course one has \psi^7=\identity,
etcetera… the group here is
essentially just integers-mod-7:
{ [0], [1], [2], [3], [4], [5], [6]},
with addition defined by
“cancelling away” multiples
of 7 (forget about the fancily
denoted “permutation structure”
displayed in defining \phi
and just look at the exponents).

the “cycle notation” used here
is *much* under-used, in my opinion.
but we’ll really only need it
for future slides.

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