cosets

March 8, 2014 in Finite Geometry, Graphics

here’s the same group…
call it G… appearing as the
lower-left hex board
from a set of four.

we intend to display an accurate
visual representation of the
simple group (S) of order 168.
our group (G) is one of many
7-element subgroups (this
particular one is selected
because \psi is easily memorized).

what we *have* displayed
on this sheet of paper is G
together with three of its
right-cosets:
{G = G1, Ga, Gb, Gc}.
it so happens that
{1, a, b, c} form a group
in their own right (a V_4
or “klein 4-group”).
however, we *don’t* get
a group structure on the cosets
out of this.

i’ve drawn one more sheet
looking quite a bit like this one
(4 more cosets of G).
when i’ve drawn four more,
that’ll be S itself (and a
colorful treat for eyes and brains).

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