Re: Sharply 5-Transitive: M12-

PaulHjelmstad wrote:
The order of M12 is 95,040, which is 132 * 720

Since M12 is the automorphism group of S(5,6,12),
and is sharply 5-transitive, and maps blocks to
blocks,
would there be 720 operations to go from say,
hexad(A)
to hexad(B), (and B to C, etc) such that

x1, x2, x3, x4, x5 -> y1, y2, y3, y4, y5; and every
combination

x1, x2, x3, x4, x5 -> (y's scrambled 5!) which
makes 120,
and then times a factor of 6, because there are 6
pentads in each hexad, such that C6,1 = C6,5 pentads,
to get every 720 g's going between two hexads? I
sense it is more complex than this. I also am
assuming all hexads are treated equally, which is
probably wrong, such that
there are 132 cycles between hexads, going
A,B,C,...
A,C,E... (and B,D,F..), A,D,G etc

Or does sharply transitive mean there is only one
(not 120) going between g1 through g5 -> h1 through
h5?

Rotman seems to be beyond my scope, even though I
understand some of the theorems and proofs for this

PGH

"Sharply transitive" means here that for each pair of
pentads P1, P2 there is exactly one
permutation in M12 that maps P1 onto P2. The pentads
are 5-element subsets with no order
relations; they are just subsets.
For a specified M12 subgroup of S12 this means that
once you have a permutation

(x1, x2, x3, x4, x5, . . . . . . . ) -> (y1, y2, y3,
y4, y5, . . . . . . .)

in which the entire x-subset is mapped onto the
entire y-subset, you cannot determine
which x_i is mapped onto which y_j.

One would expect that there exist 120 subgroups of
S12 isomorphic to M12 and all of them
mutually conjugate.

BTW, the Mathieu group M24 is transitive but not
sharply transitive on pentads within a
set of 24 elements. One would expect that within M24
indeed 120 permutations would map a
given pentad onto another given pentad. I do not know
by heart if this is true.

See ...
http://en.wikipedia.org/wiki/Transitive_group_action#T
ypes_of_actions
http://en.wikipedia.org/wiki/Mathieu_group

Happy explorations: Johan E. Mebius

Thanks. That is what I thought, they are just sets. It would be too easy to merely permute the order. By mutally conjugate I take it that all 120 subgroups are in the same conjugacy class?

It would be fun to get a literal mapping of hexads, to which hexads, but I guess then I would need a full listing of all 95,040 g's in M12! I will continue to struggle with Rotman, and if 132 hexads are
treated similarly, I guess 720 g's would explain things.

PGH
.

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