Re: Sharply 5-Transitive: M12

On 24 Jun, 18:42, PaulHjelmstad <phjelms...@xxxxxxx>
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

I have found the answers you have received to this
confusing!

Saying G acts sharply 5-transitive on a set X means
that for any
distinct x1,x2,x3,x4,x5 in X and any distinct
y1,y2,y3,y4,y5 in X,
there exists exactly one permutation g that maps
xi->yi for all i. So
the order matters here, and it means that there are
exactly 120
permutation in G that map the set {x1,x2,x3,x4,x5} to
{y1,y2,y3,y4,y5}.

So M12 sharply 5-transitive on 12 points means that
its order is
12*11*10*9*8 = 95040.

Saying a group is 5-transitive means that there is at
least
permutation with the above property - "sharply" means
exactly 1.

M24 on 24 points is 5-transitive but not sharply so.
In fact there are
48 permutations mapping any ordered pentad of
distinct points to any
other, so |M24| = 24*23*22*21*20*48 = 244823040.

Derek Holt.

Thanks Dr. Holt for your answer. I've been going over
my earlier questions on M24 (Steiner Systems), finally things are falling into place. So order does matter! I
have been studying the construction of M12 from the
inner and outer automorphisms of S6, Steven
Cullinane's Web pages have been fascinating (Finite
Geometry, Diamond Theorem). Also I should say JEMebius's
pages and applets are also pretty amazing.

PGH
.

Relevant Pages

  • Re: Sharply 5-Transitive: M12
    ... and is sharply 5-transitive, and maps blocks to blocks, ... there exists exactly one permutation g that maps xi->yi for all i. So ... So M12 sharply 5-transitive on 12 points means that its order is ...
    (sci.math)
  • Re: Sharply 5-Transitive: M12
    ... Since M12 is the automorphism group of S, ... and is sharply 5-transitive, and maps blocks to blocks, ... there exists exactly one permutation g that maps xi->yi for all i. So ...
    (sci.math)
  • Re: Sharply 5-Transitive: M12
    ... Since M12 is the automorphism group of S, ... and is sharply 5-transitive, and maps blocks to blocks, ... there exists exactly one permutation g that maps xi->yi for all i. So ...
    (sci.math)
  • Re: Sharply 5-Transitive: M12-
    ... Since M12 is the automorphism group of S, ... assuming all hexads are treated equally, ... pentads P1, ... permutation in M12 that maps P1 onto P2. ...
    (sci.math)
  • Re: permutation mappings
    ... > PI maps the set of n-bit integers into itself and no two integers map ... DES (Data Encryption Standard, for those who ... > fashion) is such a permutation for 64 bit integers. ... > mapping, then a fixed point comes points to a message that encrypts to ...
    (sci.crypt)