Posted by: **Alexandre Borovik** | October 6, 2010

## Enumeration of groups of order 720

I discovered this quotation in some research policy document:

One of the first areas of pure mathematics to be computerised was group theory, as predicted in1945 by Alan Turing who wrote of his design for the NPL ACE computer, that

*There will be positively no internal alterations to be made even if we wish suddenly to switch from calculating the energy levels of the neon atom to the enumeration of groups of order 720*.

Why there was a need for a computer to enumerate groups of order 720? Even in Turing’s times, published papers already contained all the ingredients for (admittedly, long and tedious) enumeration of groups of order 720 by hand. Although one ingredient that was missing was perhaps the Hall-Higman Theorem (not even the theorem itself, but the ideology of using linear algebra on “internal modules” in groups.

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