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Clearly this doesn't predate Cayley's discovery of the theorem or Jordan's publication of it, but it might be worth mentioning anyway. Perhaps
Burnside was unaware of Jordan's earlier publication of the theorem, it was brought to his attention after the first edition of his book, and he corrected
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That is ridiculous, the property "G is a permutation group" is true for any permutation group, but not true in general… (in the same spirit, "G is not finite, or there exists some k such that Card(G) = k!" does not hold for any group even if it holds for any permutation group).
474:(Addendum: I cannot find an online copy of the second edition of the book. The Internet Archive copies are all the 1897 edition, despite being described as being from 1911. The Project Gutenberg version is 1897 also. —
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G doesn't need to be finite - it can always be naturally embedded in its automorphism group in Set. Of course if G is infinite then that group isn't isomorphic to S_n for any finite n.
213:(i.e. U(G) or Forgetful(G)). I think that this distinction is critical to understanding this theorem since it is about a natural transformation between functors between categories
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Many of the equations in this article can be produced in LaTeX. Some of the equations might be more readable if they were converted to LaTeX from plaintext.
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There is a copy in Google Books. The page is again (§20, p. 22.) This time the theorem is stated this way:
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The title of section 20 has been changed from “Dyck's
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But does the original form of the theorem requires the group to be finite? Sorry, forgot to login.
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Knowledge. If you would like to participate, please visit the project page, where you can join
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This may be too obvious, but probably should spell it out explicitly in the introduction.
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Thus, theorems which are true for permutation groups are true for groups in general.
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The wording throughout does not always distinguish carefully between G as a
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I am not sure what to do with it, but perhaps it ought to be mentioned? —
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is capable of representation as a group of substitutions of
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38:It is of interest to the following
537:The reference given this time is:
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109:and see a list of open tasks.
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