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Next: Acknowledgments. Up: Applications. Previous: Möbius functions.

Idempotents in the Burnside ring.

We identify the Burnside ring tex2html_wrap_inline5187 with its image under the map tex2html_wrap_inline5189 in tex2html_wrap_inline5191 .

The only idempotents in tex2html_wrap_inline5193 are 0 and 1. Each idempotent in tex2html_wrap_inline5199 is of the form tex2html_wrap_inline5201 where tex2html_wrap_inline5203 or 1 for all tex2html_wrap_inline5207 . Let tex2html_wrap_inline5209 be such that its entry in th i-th position equals 1 and all other entries are 0 and, for any tex2html_wrap_inline5217 , let tex2html_wrap_inline5219 .

The table of marks M(G) provides an efficient means to determine those subsets tex2html_wrap_inline5223 for which the idempotent tex2html_wrap_inline5225 is an element of tex2html_wrap_inline5227 .

Let X be a G-set and suppose tex2html_wrap_inline5233 such that H is a normal subgroup of index p of K for some prime p. Then K acts on the fixed point set tex2html_wrap_inline5245 with kernel containing H. Since tex2html_wrap_inline5249 is prime, the orbits of K on tex2html_wrap_inline5253 have either length 1 (corresponding to tex2html_wrap_inline5257 ) or length p. It follows that

displaymath5185

in other words, if H lies in the conjugacy class tex2html_wrap_inline5267 of subgroups of G and if K lies in class tex2html_wrap_inline5273 then the i-th column and the j-th column of the table of marks M(G) are equal modulo p.

In order to describe the idempotents of tex2html_wrap_inline5283 we define two relations on the set tex2html_wrap_inline5285 . For any prime p let

tex2html_wrap_inline5289
if there are tex2html_wrap_inline5291 such that H is normal of index p in K, H lies in tex2html_wrap_inline5301 and K lies in tex2html_wrap_inline5305 ;
tex2html_wrap_inline5307
if the i-th and the j-th column of M(G) are equal modulo p;

The idempotents of tex2html_wrap_inline5317 are then completely described by the following proposition (see [Dress 1969], Proposition 1).

Prop1014

In particular, it is possible to decide from the table of marks of G whether G is solvable or not:

Since solvable groups are characterized by the fact that every nontrivial subgroup has a normal subgroup of index p for some prime p we get the following characterization of solvable groups.

Thm1018


next up previous
Next: Acknowledgments. Up: Applications. Previous: Möbius functions.

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