Re: Sharply 5-Transitive: M12

Derek Holt wrote:
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.

My thanks for recalling the definition of transitivity - in my newsgroup post of yesterday I was mistaken. Once in a while I simply forget to check things I learned in my student's years in the 1960s.
Robert D. Carmichael's "Introduction to the theory of groups of finite order" (*) is my favourite reference to finite groups.

(*) Copyright 1937 by Robert D. Carmichael. Unabridged and unaltered republication 1956 by Dover Publications, Inc.

Johan E. Mebius
.

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