[GAP Forum] Metrics for large and small permutation groups.

[Jonathan Cohen]

Hi Forum,

Let me try and clear up some of Derek Holt's concerns, now that I am over
my "holiday cheer" :)

> You have just said that "large base" applies to families of permutation groups
> rather than to individual groups,

Yes, sorry, I had slipped into abusing the definition already.

>so presumably you mean that the family of
> symmetric groups { S_n | n in N } *is* small base.

It is large base.

> > -- The cyclic groups are small base, since every cyclic groups admits
> > a base of length 1
>
> You have not specified a family of permutation groups here.
> The cyclic group generated by (1,2)(3,4,5) has no base of length 1 .

Yes, a silly slip on my part, I should have specified the cyclic groups
that act regularly (or all semiregular groups for that matter).

> > -- The non-alternating finite simple groups are small base (this
> > follows from a result of cameron)
>
> The non-alternating finite simple groups form a family of (isomorphism
> classes of) abstract groups rather than a family of permutation groups.
> Presumably you are saying here that the family of all equivalence classes of
> permutation representations of all non-alternating finite simple groups
> is a small base family?

I mean the family of all permutation representations of all
non-alternating finite simple groups. I am not quite sure how you are
taking the equivalence classes. The reading {G | G is a permutation
representation of H} could lead to trouble, since the minimal base
length of a member of the class will most likely vary according to the
representation.

> > The *primitive* large base groups are the subgroups of wreath products
> > S_n wr S_m which contain (A_n)^m, where the wreath product is in its
> > product action and the action of S_n is on k-element subsets of
> > {1,2,...,n} - this follows from cameron's result.
>
> Again, it is not immediately clear how to formulate the above statement
> in terms of families of groups.

One way is to generate lots of families by fixing k and letting n and m
vary (so one family for each choice of k). One can show that such a family
is large base. However, I cannot see how to do it without fixing k since,
in that case, we may pick (for example) the subfamily where k=n-1.

I omitted an interesting sidenote in my previous message. All of the cited
results require CFSG, however, Pyber provided a completely elementary proof for
the case of doubly transitive permutation groups. His paper is:

L. Pyber. On the orders of doubly transitive permutation groups,
elementary estimates. J. Combin. Theory Ser. A, 62(2):361-366, 1993.

Jon

--
http://www.maths.uwa.edu.au/~cohenj02