Comments on: Permutation groups and regular semigroups

By: Peter Cameron

Peter Cameron — Thu, 15 Dec 2011 19:52:19 +0000

Yes, that's right. I used to use this term but now one of the things I work on is "homogeneous relational structures" where the word has a different meaning. Terminology is difficult. For relational structures, people used to use "ultrahomogeneous" and "homogeneous", roughly parallel to "transitive" and "set-transitive" for groups. Then, when they settled on "homogeneous" for the stronger concept, at least one researcher started using "transitive" for the weaker one. [In the stronger concept, whenever you have an isomorphism between finite substructures, you can extend it to an automorphism of the whole structure; in the weaker concept, whenever two structures are isomorphic, then some automorphism of the whole structure carries one to the other. These exactly agree with the permutation concept if "structures" are k-element subsets and "automorphisms" are elements of the group.]

By: Bill Fahle

Bill Fahle — Thu, 15 Dec 2011 14:29:31 +0000

Oh, right. I think they call that k-homogenous in the book I’m looking at: Finite Groups III by Huppert and Blackburn. Thanks for your response.

By: Peter Cameron

Peter Cameron — Wed, 14 Dec 2011 19:54:10 +0000

Sorry, my mistake: it should be set-transitivity. I’ll fix it.

By: Bill Fahle

Bill Fahle — Wed, 14 Dec 2011 14:49:57 +0000

I realize this post is a few months old, but I have recently become interested in k-transitive groups, particularly Mathieu groups. You said that “k-transitivity is equivalent to n−k-transitivity for a permutation group on n points. So, without loss, we may assume that k≤n/2.” But it is known that, for example, that M12 is 5-transitive on 12 points. But it is not also 7-transitive, is it? That would contradict your other statement that “In particular, the only k-transitive groups for k≥6 are the symmetric and alternating groups. (There is no prospect of a proof without CFSG.)” I appreciate your response.

By: Benjamin Steinberg

Benjamin Steinberg — Sat, 08 Oct 2011 03:04:54 +0000

Peter, Mohan Putcha takes this view for the representation theory of monoids, but perhaps with the opposite goal. He asks how does an irreducible representation of a monoid decompose as a representation of its unit group. For any finite group G of Lie Type he and Renner constructed a canonical monoid of Lie type M with unit group G. The Steinberg representation is constructed as the restriction of an irreducible representation of the monoid to the group.

Putcha also studied how semi simplicity of the algebra of invariants for the action of the unit group can in important cases imply semi simplicity of the monoid algebra.

Monoids if Lie type have essentially Zariski dense groups of units and so they control the monoid structure. In general the group of units can be too small to influence the monoid.