finally added pointer to

- Tammo tom Dieck, Chapter IV of:
*Transformation Groups*, de Gruyter 1987 (doi:10.1515/9783110858372)

Will give this book its own `category:reference`

-entry, to make fully transparent that this and *Transformation Groups and Representation Theory* are two distinct books, albeit by the same author…

Thanks!

Interesting: I find the sections 184-185 in that 1911 textbook version considerably less clear than the 1901 article. Not easy to spot the concept of the Burnside ring here, even if one knows what one is looking for and where to look for it.

Anyway, so I’ll be citing the 1901 article. That’s what I wanted to know.

]]>I found a copy so replaced the link

- {#Burnside1897} William Burnside,
*Theory of Groups of Finite Order*, Second edition Cambridge 1911 (pdf)

Okay, I have now expanded the beginning of the References-section (here) to read as follows:

The Burnside product seems to first appear as equation (i) in:

- {#Burnside01} William Burnside,
*On the Representation of a Group of Finite Order as a Permutation Group, and on the Composition of Permutation Groups*, Proceedings of the American Mathematical Society 1901 (doi:10.1112/plms/s1-34.1.159)

(beware the terminology: a G-set is called a “permutation group $G$” in that article, a subset is called a “compound” and the Cartesian product of $G$-sets is called their “compounding”).

It is then included (not in the first but) in the second edition (Sections 184-185) of:

{#Burnside1897} William Burnside,

*Theory of Groups of Finite Order*, Second edition Cambridge 1911 (first edition pdf)reprinted by Cambridge University Press 2012 (doi:10.1017/CBO9781139237253)

The term “Burnside ring” as well as “Burnside algebra” is then due to (see NMT 04, Vol. 1 p. 60 for historical comments)

- {#Solomon67} Louis Solomon,
*The Burnside algebra of a finite group*, Journal of Combinatorial Theory Volume 2, Issue 4, June 1967, Pages 603-615 (doi:10.1016/S0021-9800(67)80064-4)

Excellent, thanks! Am editing this into the entry now..

Coming back to that book: our pdf link seems to be to the first edition then, since Sections 184-185 there are about polygons. Do we have a pdf-copy of that second edition (or later)?

]]>It looks like it is Solomon then who names it in 1967. Hazelwink claims it’s Dress (“According to some the Burnside ring was introduced by Andreas Dress in [117]”), but that’s to a 1969 paper, and Dress doesn’t call it the Burnside ring there anyway.

]]>Thanks again.

Since that second edition of the book is from 1911, the earliest reference might indeed be that other article you found:

- William Burnside,
*On the Representation of a Group of Finite Order as a Permutation Group, and on the Composition of Permutation Groups*, Proceedings of the American Mathematical Society 1901 (doi:10.1112/plms/s1-34.1.159)

where the Burnside product appears as equation (i).

After doing some fun translation:

his “permutation group $G$” is our “$G$-set” (!),

his “compound” is our “sub-set”

his “compounding” is our “Cartesian product”.

In The Burnside algebra of a finite group, Solomon writes

The isomorphism classes of $G$-sets may be added and multiplied in natural fashion and generate a commutative ring $\mathcal{B}[G]$ which, since it seems to have been defined for the first time in Burnside’s book [3, Secs. 184-5], we call the Burnside ring of $G$.

That’s the second edition importantly denoted [3].

]]>Hm, in rev 12 I had added a line saying

The concept was named by Dress, following [ Burnside 1897 ]

I must have read this somewhere, but I forget where.

]]>Thanks!!

I’ll look into it in a moment…

]]>Some commentary here, which claims Solomon coined the name.

]]>According to the commentary in his collected papers, the origins are to be found in this 1901 paper, On the Representation of a Group of Finite Order as a Permutation Group, and on the Composition of Permutation Groups, and then sections 184-185 of the second edition (1911) of his monograph.

]]>I’d like to cite the invention of the Burnside ring, in whatever guise. Is there a page in that book on which we can recognize the concept being conceived?

]]>I doubt anyone this early is calling anything a ring other than collections of numbers or polynomial. Abstract rings certainly come later. Or do you just mean where he talks about something we’d recognise as a ring?

]]>Hm, where exactly in Burnside’s original book does he actually introduce the Burnside ring? Or does he even?

]]>added re-publication data to:

- William Burnside,
*Theory of Groups of Finite Order*, 1897 (pdf), reprinted by Cambridge University Press 2012 (doi:10.1017/CBO9781139237253

Nice! Tweaked slightly to make the statement hopefully completely clear.

]]>added a Properties-section (here) on expressing the Burnside product in terms of the table of marks

]]>added pointer to

- Serge Bouc,
*Burnside rings*, in*Handbook of Algebra*Volume 2, 2000, Pages 739-804 (doi:10.1016/S1570-7954(00)80043-1)

Thanks for the pointer.

He attributes the observation that the Barratt–Priddy–Quillen theorem may be read as sying $\mathbb{S} \simeq K \mathbb{F}_1$ to

- Y. Manin.
*Lectures on zeta functions and motives (according to Deninger and Kurokawa)*. Ast ́erisque, (228):4, 121–163, 1995. Columbia University Number Theory Seminar (New York, 1992) (pdf)

I have looked through that pdf. While I see it talk about $\mathbb{F}_1$, I didn’t find a remark yet concerning Barratt–Priddy–Quillen…

]]>There might be some interesting ideas about $G$-equivariant $\mathbb{F}_1$-theory in

- Snigdhayan Mahanta,
*$G$-theory of $\mathbb{F}_1$-algebras I: the equivariant Nishida problem*, (arXiv:1110.6001)

added remark that Segal’s theorem $A(G) \overset{\simeq}{\longrightarrow} \mathbb{S}_G(\ast)$ is a special case of the tom Dieck splitting theorem

]]>I believe the initial notion of $\beta$-ring is, in fact, due to another ex-colleague! (Remember Dudley Littlewood was Ronnie Brown’s predecessor at Bangor and he was central in the development of permutation representation theory.) I really should look back over that stuff, and will if I have a moment. Guillot’s paper looks interesting. Thanks.

]]>The point in Guillot 06 “Adams operations in cohomotopy” is to show that it is not quite a $\lambda$-ring, but a “$\beta$-ring”.

]]>