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The Burnside Ring and the Table of Marks.

 

In this section we recall basic facts about finite group actions. The table of marks is defined and its relation to the subgroup structure and to the Burnside ring is described.

Let G be a finite group. Denote by tex2html_wrap_inline2259 the set of all subgroups of G. Then tex2html_wrap_inline2263 is a partially ordered set (poset) with incidence relation tex2html_wrap_inline2265 . The group G acts on tex2html_wrap_inline2269 by conjugation, i.e. via tex2html_wrap_inline2271 for tex2html_wrap_inline2273 and tex2html_wrap_inline2275 . This G-action respects incidence: if tex2html_wrap_inline2279 then tex2html_wrap_inline2281 for all tex2html_wrap_inline2283 and tex2html_wrap_inline2285 . We denote the G-orbit of tex2html_wrap_inline2289 (i.e. the conjugacy class of subgroups of G which contains U) by tex2html_wrap_inline2295 and usually omit the subscript as long as no confusion can arise. The set of G-orbits tex2html_wrap_inline2299 also is a poset, with incidence tex2html_wrap_inline2301 if tex2html_wrap_inline2303 for tex2html_wrap_inline2305 and some tex2html_wrap_inline2307 . We will refer to tex2html_wrap_inline2309 as the poset structure of G.

A (right) G-set X is a set X together with an action tex2html_wrap_inline2319 such that tex2html_wrap_inline2321 and tex2html_wrap_inline2323 for all tex2html_wrap_inline2325 and all tex2html_wrap_inline2327 . Every G-set X decomposes into a disjoint union of orbits tex2html_wrap_inline2333 , each of which is a G-set itself. A G-set is transitive if it consists of only one orbit. All G-sets in this article are assumed to be finite.

A homomorphism between two G-sets X and Y is a map tex2html_wrap_inline2347 such that tex2html_wrap_inline2349 for all tex2html_wrap_inline2351 and all tex2html_wrap_inline2353 . Two G-sets X and Y are isomorphic if there exists a bijective homomorphism tex2html_wrap_inline2361 .

Let tex2html_wrap_inline2363 be representatives of the conjugacy classes of subgroups of G. Then tex2html_wrap_inline2367 . For each subgroup tex2html_wrap_inline2369 the group G acts transitively on the set tex2html_wrap_inline2373 of right cosets of U in G. Conversely every transitive G-set X is isomorphic to a G-set tex2html_wrap_inline2385 where U is a point stabilizer of X in G. For every tex2html_wrap_inline2393 the G-set tex2html_wrap_inline2397 is isomorphic to tex2html_wrap_inline2399 . Thus every transitive G-set is isomorphic to tex2html_wrap_inline2403 for some tex2html_wrap_inline2405 .

Def139

Rem152

Let tex2html_wrap_inline2455 and consider the G-set tex2html_wrap_inline2459 . Then U has fixed points in that action if and only if U is contained in a one point stabilizer, i.e. in at least one conjugate of tex2html_wrap_inline2465 . Thus the table of marks describes the poset tex2html_wrap_inline2467 : the incidence matrix of this poset is obtained from M(G) by replacing every nonzero entry by 1.

But M(G) contains far more information about the subgroup structure of G. This is due to the following recalculation of the value of a mark.

Prop156

proof162

The following lemma collects some easy consequences of the above formula. In particular the numbers of incidences between two conjugacy classes of subgroups of G can be derived from M(G).

The table of marks of the alternating group tex2html_wrap_inline2511 of order 60 in Table 1 serves as an example. tex2html_wrap_inline2515 has nine conjugacy classes of subgroups. They are distinguished by their orders and have isomorphism types: 1, 2, 3, tex2html_wrap_inline2523 , 5, tex2html_wrap_inline2527 , tex2html_wrap_inline2529 , tex2html_wrap_inline2531 , and tex2html_wrap_inline2533 . The rows of the table correspond to the transitive G-sets tex2html_wrap_inline2537 .

   table180
Table 1: The table of marks of tex2html_wrap_inline2665 .

  La216

Denote by tex2html_wrap_inline2681 the number of conjugates of a subgroup U of G contained in a fixed subgroup V of G. These numbers also are determined by M(G).

  Prop239

proof254

On the other hand M(G) is determined by the numbers tex2html_wrap_inline2713 for all tex2html_wrap_inline2715 and the additional knowledge of the index tex2html_wrap_inline2717 for every tex2html_wrap_inline2719 .

  La262

Denote for any G-set X its isomorphism class by [X]. The Burnside ring tex2html_wrap_inline2731 of G is the free abelian group

displaymath2232

generated by the isomorphism classes of transitive G-sets tex2html_wrap_inline2737 , tex2html_wrap_inline2739 . Here the sum [X] + [Y] of the isomorphism classes of G-sets X and Y is the isomorphism class tex2html_wrap_inline2749 of the disjoint union of X and Y. Moreover, their product tex2html_wrap_inline2755 is the isomorphism class tex2html_wrap_inline2757 of the Cartesian product of X and Y. This turns tex2html_wrap_inline2763 into a commutative ring with identity tex2html_wrap_inline2765 .

Let X and Y be G-sets and let tex2html_wrap_inline2773 . Then

displaymath2233

Therefore

displaymath2234

Thus, if we define tex2html_wrap_inline2775 for each tex2html_wrap_inline2777 to be the r-tuple tex2html_wrap_inline2781 , then the map

displaymath2235

is a ring homomorphism from tex2html_wrap_inline2783 to tex2html_wrap_inline2785 .

Let tex2html_wrap_inline2787 . Then tex2html_wrap_inline2789 can be expressed in terms of the table of marks M(G) as

displaymath2236

Moreover, the G-set X is characterized up to isomorphism by tex2html_wrap_inline2797 .

  Thm284

proof287

Let X be a G-set. The permutation character tex2html_wrap_inline2829 of G on X is defined as tex2html_wrap_inline2835 for any element tex2html_wrap_inline2837 . This number, of course, coincides with the mark tex2html_wrap_inline2839 of the cyclic subgroup generated by g on X. Therefore, the table of marks M(G) contains in the columns corresponding to cyclic subgroups a complete list of transitive permutation characters tex2html_wrap_inline2847 of G corresponding to the transitive G-sets tex2html_wrap_inline2853 .

The following proposition (see [Kerber 1991], 3.2.18) provides a way to determine the columns of M(G) which correspond to cyclic subgroups.

Let tex2html_wrap_inline2857 be the primitive idempotent of tex2html_wrap_inline2859 corresponding to the conjugacy class of tex2html_wrap_inline2861 , that is tex2html_wrap_inline2863 , and write tex2html_wrap_inline2865 with rational coefficients tex2html_wrap_inline2867 . (The matrix tex2html_wrap_inline2869 then is the inverse of M(G).)

Prop304

proof315


next up previous
Next: Induction of Marks. Up: The Subgroups of Previous: The Subgroups of

Götz Pfeiffer Wed Oct 30 09:52:08 GMT 1996