# Re: paper claiming projective planes must have prime power order

*From*: Craig Feinstein <cafeinst@xxxxxxx>*Date*: Mon, 16 Feb 2009 08:02:24 -0800 (PST)

On Feb 14, 9:42 pm, Craig Feinstein <cafei...@xxxxxxx> wrote:

On Feb 12, 10:44 pm, Craig Feinstein <cafei...@xxxxxxx> wrote:

For people who don’t have time to read Mehendale’s whole paper, I’ll

summarize his argument in the paper arXiv:math/0611492, which claims

that a finite projective plane must have prime power order:

The author proves that if there are n-1 orthogonal Latin squares of

order n, then n must be a prime power. (Bose’s Theorem tells us that

this is equivalent to proving that a finite projective plane must have

prime power order.)

Consider the n(n-1) rows of n-1 Latin squares of order n as a set of

permutations in the group of permutations S_n. For instance if n=5 and

a row is 3 2 4 1 5, then this means that 1 goes to 3, 2 goes to 2, 3

goes to 4, 4 goes to 1 and 5 goes to 5. Without loss of generality,

assume that one of the Latin squares has the identity permutation [1 2

… n] as a row. Call the set of rows in this Latin square N. And

without loss of generality, assume that the first column in each of

the Latin squares is [1 2 … n]. Call the set of n(n-1) rows G.

Then it is not difficult to see that any two columns of G (when all of

the rows are placed on top of one another) must contain the n(n-1)

unique ordered pairs (1,2),(1,3),…,(1,n),(2,1),(2,3),….,(n,1),(n,2),…,

(n,n-1). (Otherwise, the n-1 Latin squares would not be orthogonal.)

In other words, the sets of permutations must be sharply 2-transitive.

This implies that the set G of permutations is a group, since there is

an identity permutation in G.

When I wrote this, I thought that a sharply 2-transitive set of

permutations must be a group. Now, I'm not so certain. I wrote the

author of the paper asking him for a proof of this.

After contacting the author of the paper, it is clear to me that I

misunderstood his paper. By mistake, I misrepresented what the author

had in mind when I summarized his paper in this thread. I apologize

for this.

In the author's words to me through email, he agreed that "There can

exist more than one nonisomorphic projective planes of given order,

for which each one may not admit group structure." So the reference to

"exotic" projective planes of order 9 does not contradict what

Mehendale had in mind when he was writing this paper. It only

contradicts my former interpretation of what Mehendale's paper, which

was wrong.

I'm going to leave this thread and leave it to others to try to figure

this all out.

Craig

Furthermore, the set N consists of the identity permutation and n-1

permutations in G that are fixed-point-free, since these permutations

form a Latin square. (One can use an elementary counting-argument to

show that there are at most n-1 permutations in G that are fixed-point-

free, since G is sharply 2-transitive.) Since N is sharply transitive,

N is a subgroup of G. Also, N is a normal subgroup in G: A conjugate

of any element in N must also be fixed-point-free. But since the only

permutations in G that are fixed-point-free are in N, the nonidentity

elements of N must all be conjugate to one another. Hence, the

nonidentity elements of N must have the same order, some prime p

divisible by n. If n is not a power of the prime p, then by Cauchy’s

theorem N must have an element of order not equal to p and not equal

to 1, which is impossible, since all elements of N have order either p

or 1. Hence, n must be a prime power.

Notice that this argument doesn’t require the theorem that Mehendale

quotes from Aschbacher’s book. You just need Cauchy’s theorem for

this.

There you have it - an elementary proof of a famous conjecture that

has been open for a very long time and that has been up on arxiv.org

for more than two years now, yet the conjecture is still considered by

an open problem by the mathematics community at large. Can anyone find

a flaw in this argument? I can’t.- Hide quoted text -

- Show quoted text -- Hide quoted text -

- Show quoted text -

.

**Follow-Ups**:**Re: paper claiming projective planes must have prime power order***From:*victor_meldrew_666@xxxxxxxxxxx

**Re: paper claiming projective planes must have prime power order***From:*hagman

**References**:**paper claiming projective planes must have prime power order***From:*Craig Feinstein

**Re: paper claiming projective planes must have prime power order***From:*Craig Feinstein

**Re: paper claiming projective planes must have prime power order***From:*Craig Feinstein

**Re: paper claiming projective planes must have prime power order***From:*Craig Feinstein

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