
The ‘Useless’ Perspective That Transformed Mathematics - theafh
https://www.quantamagazine.org/the-useless-perspective-that-transformed-mathematics-20200609/
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mathgenius
From TWF #252 [1]:

""" Back in 1897, a mathematician named William Burnside wrote the first book
in English on finite groups. It was called Theory of Groups of Finite Order.

In the preface, Burnside explained why he studied finite groups by letting
them act as permutations of sets, but not as linear transformations of vector
spaces:

'Cayley's dictum that "a group is defined by means of the laws of combination
of its symbols" would imply that, in dealing with the theory of groups, no
more concrete mode of representation should be used than is absolutely
necessary. It may then be asked why, in a book that professes to leave all
applications to one side, a considerable space is devoted to substitution
groups [permutation groups], but other particular modes of representation,
such as groups of linear transformations, are not even referred to. My answer
to this question is that while, in the present state of our knowledge, many
results in the pure theory are arrived at most readily by dealing with
properties of substitution groups, it would be difficult to find a result that
could most directly be obtained by the consideration of groups of linear
transformations.'

In short, he didn't see the point of representing groups on vector spaces - at
least as a tool in the "pure" theory of finite groups, as opposed to their
applications.

However, within months after this book was published, he discovered the work
of Georg Frobenius, who used linear algebra very effectively to study groups!

So, Burnside started using linear algebra in his own work on finite groups,
and by the time he wrote the second edition of his book in 1911, he'd changed
his tune completely:

Very considerable advances in the theory of groups of finite order have been
made since the appearance of the first edition of this book. In particular the
theory of groups of linear substitutions has been the subject of numerous and
important investigations by several writers; and the reason given in the
original preface for omitting any account of it no longer holds good. In fact
it is now more true to say that for further advances in the abstract theory
one must look largely to the representation of a group as a group of linear
transformations. It's interesting to see exactly how representing finite
groups on vector spaces lets us understand them better. By now almost everyone
agrees this is true. But how much of the detailed machinery of linear algebra
is really needed? """

[1]
[http://math.ucr.edu/home/baez/week252.html](http://math.ucr.edu/home/baez/week252.html)

