15 April, 2008

Hawking + Galois ≠ Normal Group

I took my son to Books-A-Million some time ago and saw that Steven Hawking has a bunch of books out. In the book God created the integers...* he has selected excerpts from some of the greatest advances in mathematics. From Εὐκλείδης (Euclid) to Gauß (Gauss) to Лобачевский (Lobachevsky) to Galois, Hawking discusses briefly a biography of each mathematician, the contribution made, and the importance of that contribution. I browsed it and was impressed.

If you look for it online, you will find a lot of negative reviews. Why? Many people complain about typos and errors. I hate to say this, but they have a point. Hawking sometimes gets his math wrong, seriously wrong.

The most egregious example I found was in Galois theory. Here is how Hawking summarizes it on page 805 of the text:

To be brief, Galois demonstrated that the general polynomial of degree n could be solved by radicals if and only if every subgroup N of the group of permutations Sn is a normal subgroup. Then he demonstrated that every subgroup of Sn is normal for all n ≤ 4 but not for any n > 5.
Unfortunately, Galois Theory cannot be summarized so briefly. Even a reader who is not mathematically inclined may be wondering, What about the case where n = 5? A typographical error has concealed the fact that the case n = 5 falls into the latter category (of general polynomials whose roots cannot be found by radicals).

That typo pales in comparison to a bigger problem, which is that Hawking's explanation is wrong. Not every subgroup of S3 is normal: for example, {(1),(12)} is not normal. Nor is every subgroup of S4 normal. Hawking's statement implies that the general polynomials of degree 3 and 4 cannot be solved by radicals. This is false.

What actually happens is that you have to construct a certain sequence of "proper normal subgroups". This becomes impossible for n ≥ 5 because a normal subgroup of Sn called An has no proper normal subgroups, and we need it to have them.

This is a strange error to have in such a book. I wonder if it's merely an example of how even great scientists can make some pretty bad mistakes when they try to summarize the work of other great scientists outside their own fields.

I am obviously not a great scientist; I make mistakes when summarizing my own work. Just this past Friday, I worked myself into an obsessive panic after thinking a theorem I wrote a year and a half ago is "obviously wrong". I didn't realize that it was correct after all until that evening while I was putting my oldest daughter to sleep. That mistake was amazingly important nevertheless; it provided me with an insight that I hadn't observed a year and a half ago—or if I did, I had forgotten it.

All that said, Hawking is right; these are some of the most beautiful and most important results in mathematics. If you have the time to peruse it and the energy to get around the errors and typos, you might find it a salutary read.

*"God created the integers...": This title is due to a quote of Leopold Kronecker, who was quite a character in mathematics. The full quote is "God created the integers; all else is the work of man."

[This post was 3 months in the making; the original draft was on 13 January 2008. That's how long it took me to get around to returning to Books-A-Million, finding the book in question, and verifying the quote above. I didn't want to write this sort of entry based on a faulty reading of the text!]

1 comment:

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