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Next: Approximating the fusion map. Up: The Subgroups of Previous: Induction of Marks.

tex2html_wrap_inline3061 , a small example.

 

Consider the projective special linear group tex2html_wrap_inline3063 of order 168. It has three conjugacy classes of maximal subgroups, two classes of type tex2html_wrap_inline3067 and one class of type 7:3 (see e.g. [Conway et al. 1985], p. 3). Assume that we already know the tables of marks of these groups, that is the corresponding posets of conjugacy classes of subgroups together with additional incidence information. The poset structure of tex2html_wrap_inline3071 is given in Figure 1 and the underlying set of elements is:

   figure391
Figure 1: The poset structures of tex2html_wrap_inline3103 and 7:3.

displaymath3057

where 2a and tex2html_wrap_inline3109 are contained in tex2html_wrap_inline3111 . The poset structure of 7:3 is also given in Figure 1 with underlying set of elements:

displaymath3058

In order to construct the poset structure of tex2html_wrap_inline3115 we start with the disjoint union of two copies of the poset structure of tex2html_wrap_inline3117 , a yellow (Y) and a red (R) one, say, and a blue (B) copy of the poset structure of 7:3. They form one colored diagram with vertex set

tex2html_wrap_inline3127 1(Y), 2a(Y), 2b(Y), 3(Y), 4(Y), tex2html_wrap_inline3139 , tex2html_wrap_inline3141 , tex2html_wrap_inline3143 , tex2html_wrap_inline3145 , tex2html_wrap_inline3147 , tex2html_wrap_inline3149 , 1(R), 2a(R), 2b(R), 3(R), 4(R), tex2html_wrap_inline3161 , tex2html_wrap_inline3163 , tex2html_wrap_inline3165 , tex2html_wrap_inline3167 , tex2html_wrap_inline3169 , tex2html_wrap_inline3171 , 1(B), 3(B), 7(B), tex2html_wrap_inline3179 .

Here a symbol like tex2html_wrap_inline3181 is just a name for a vertex in the yellow part of the disjoint union of diagrams. The fact that it denotes an elementary abelian group of order 4 follows from the information that is stored in the table of marks, rather than from its name. Of course, the names we are working with here were carefully chosen as to indicate the type of object they denote.

According to Lemma 1.3 (i) the table of marks contains information about the size of each subgroup, and conjugate subgroups of tex2html_wrap_inline3185 have the same size. Therefore, if we split the whole set of vertices into subsets according to the orders of the corresponding subgroups, then only vertices in the same subset can be conjugate in tex2html_wrap_inline3187 . This yields the following partition on the set of vertices.

tex2html_wrap_inline3189 , tex2html_wrap_inline3191 , tex2html_wrap_inline3193 , tex2html_wrap_inline3195 , tex2html_wrap_inline3197 , tex2html_wrap_inline3199 , tex2html_wrap_inline3201 , tex2html_wrap_inline3203 , tex2html_wrap_inline3205 , tex2html_wrap_inline3207 .

Our aim now is to manipulate this colored diagram with the partition of the vertices step by step until it represents the poset structure of tex2html_wrap_inline3209 . This is achieved by two sorts of manipulations:

  1. we will split a part of the partition of the vertices into subsets whenever we can ensure that only vertices lying in the same subset correspond to subgroups of tex2html_wrap_inline3211 which can possibly be conjugate in tex2html_wrap_inline3213 ,
  2. we will fuse (identify) two vertices whenever we find out that they correspond to subgroups of tex2html_wrap_inline3215 which are conjugate.
Eventually each subset will consist of only a single element: then we are done! The interested reader is encouraged to illustrate the progress we make by drawing updated versions of the colored diagram after every single manipulation.

The subsets tex2html_wrap_inline3217 and tex2html_wrap_inline3219 both contain only one element, hence each of them already determines a unique conjugacy class of subgroups of tex2html_wrap_inline3221 .

There is, of course, only one trivial subgroup in tex2html_wrap_inline3223 , so we are allowed to fuse the vertices 1(R), 1(Y) and 1(B) corresponding to the trivial subgroups of the three maximal subgroups into a single vertex which we simply denote by 1.

From the table of marks of tex2html_wrap_inline3233 we can derive that subgroups named 4 contain one cyclic subgroup of order 2, in contrast to the subgroups named tex2html_wrap_inline3239 which contain three cyclic subgroups of order 2 each. Hence these subgroups can not be conjugate in tex2html_wrap_inline3243 and we are allowed to split the set of groups of order 4 into two subsets accordingly. Now the situation is as follows.

tex2html_wrap_inline3247 , tex2html_wrap_inline3249 , tex2html_wrap_inline3251 , tex2html_wrap_inline3253 , tex2html_wrap_inline3255 , tex2html_wrap_inline3257 , tex2html_wrap_inline3259 , tex2html_wrap_inline3261 , tex2html_wrap_inline3263 , tex2html_wrap_inline3265 , tex2html_wrap_inline3267 .

By Sylow's theorem, G acts transitively on the set of Sylow p-subgroups of G for any prime p. Hence we can fuse all the subgroups of type tex2html_wrap_inline3277 (for p = 2) and all the subgroups of type 3 (for p = 3).

The normalizer in G of a group of type 3 is a group of type tex2html_wrap_inline3289 . (Note that this can also be decided from the tables of marks, since a cyclic 3 has index two in tex2html_wrap_inline3293 and the order of its normalizer in tex2html_wrap_inline3295 is 6.) Together with the subgroups of type 3 their normalizers are conjugate in G, hence we can fuse the subset corresponding to these groups into a single vertex. (See Corollary 4.10 for how knowledge about normalizers can be exploited in the general case.)

Within the Sylow 2-subgroup of type tex2html_wrap_inline3305 of tex2html_wrap_inline3307 there is only one tex2html_wrap_inline3309 -class of subgroups of type 4. Hence, by Sylow's theorem, all subgroups of this type are conjugate in G and we can fuse the yellow and the red group named 4. (See Corollary 4.8 for how knowledge about the Sylow subgroups can be exploited in the general case.)

From the character table of tex2html_wrap_inline3317 we can read off that there is only one class of elements (and hence of subgroups) of order 2, so we can fuse all vertices named 2a or 2b into a single vertex 2.

We are left with three subsets containing more than one element: those corresponding to the subgroups of type tex2html_wrap_inline3327 , tex2html_wrap_inline3329 or tex2html_wrap_inline3331 . In order to show that subgroups of type tex2html_wrap_inline3333 which are not conjugate in a maximal tex2html_wrap_inline3335 are not conjugate in G either, we examine the permutation character of G on tex2html_wrap_inline3341 . The character table of tex2html_wrap_inline3343 admits only one nontrivial character tex2html_wrap_inline3345 of degree 7 that satisfies tex2html_wrap_inline3349 for all tex2html_wrap_inline3351 , therefore tex2html_wrap_inline3353 is the permutation character of G on tex2html_wrap_inline3357 . The restriction of tex2html_wrap_inline3359 to tex2html_wrap_inline3361 admits two different decompositions into transitive components, corresponding to the two classes of maximal subgroups of type tex2html_wrap_inline3363 in G. The corresponding sums of rows of the table of marks of tex2html_wrap_inline3367 reveal different values of fixed points for the two different conjugacy classes of subgroups of type tex2html_wrap_inline3369 inside tex2html_wrap_inline3371 .

Hence the set containing the groups of this type must split in two subsets. One of them contains the red tex2html_wrap_inline3373 and the yellow tex2html_wrap_inline3375 (remember that the labeling was chosen in such a way that groups of type tex2html_wrap_inline3377 are normal in tex2html_wrap_inline3379 ). And both subsets now correspond to one conjugacy class of subgroups of G.

   figure474
Figure 2: The poset structure of tex2html_wrap_inline3413 .

Now the red tex2html_wrap_inline3415 and the yellow tex2html_wrap_inline3417 lie above different classes of groups of type tex2html_wrap_inline3419 , whence they must correspond to different classes of subgroups of G. So we split the subset tex2html_wrap_inline3423 in two parts. The same holds for tex2html_wrap_inline3425 (well, we knew right from the start that there are two classes of them in tex2html_wrap_inline3427 ) and we split the subset tex2html_wrap_inline3429 into two parts. (See Corollary 4.5 for how knowledge about numbers of subgroups can be exploited in the general case.)

Now every subset corresponds to exactly one conjugacy class of subgroups of G. We finally add the group itself. Thus we have constructed the complete poset structure of tex2html_wrap_inline3433 (see Figure 2), with vertex set

tex2html_wrap_inline3435 , tex2html_wrap_inline3437 , tex2html_wrap_inline3439 , tex2html_wrap_inline3441 , tex2html_wrap_inline3443 , tex2html_wrap_inline3445 , tex2html_wrap_inline3447 , tex2html_wrap_inline3449 , tex2html_wrap_inline3451 , tex2html_wrap_inline3453 , tex2html_wrap_inline3455 , tex2html_wrap_inline3457 , tex2html_wrap_inline3459 , tex2html_wrap_inline3461 , tex2html_wrap_inline3463 .

This small example illustrates several aspects of the general procedure. We have seen essentially two types of steps in the development of the poset of tex2html_wrap_inline3465 . Most of the conclusions, like those using Sylow's theorem or the conjugacy of normalizers, were based on general facts about the structure of finite groups. Other conclusions, like the existence of exactly one conjugacy class of subgroups of order 2 or the fusion of the subgroups of type tex2html_wrap_inline3469 , arose from additional knowledge about the particular group tex2html_wrap_inline3471 . The next section formalizes the general setting.


next up previous
Next: Approximating the fusion map. Up: The Subgroups of Previous: Induction of Marks.

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