Download The subgroup structure of the finite classical groups by Peter B. Kleidman, Martin W. Liebeck PDF
By Peter B. Kleidman, Martin W. Liebeck
With the category of the finite basic teams entire, a lot paintings has long past into the research of maximal subgroups of virtually basic teams. during this quantity the authors examine the maximal subgroups of the finite classical teams and current learn into those teams in addition to proving many new effects. particularly, the authors advance a unified therapy of the speculation of the 'geometric subgroups' of the classical teams, brought through Aschbacher, they usually solution the questions of maximality and conjugacy and procure the appropriate shapes of those teams. either authors are specialists within the box and the e-book may be of substantial price not just to crew theorists, but in addition to combinatorialists and geometers drawn to those suggestions and effects. Graduate scholars will locate it a really readable creation to the subject and it'll convey them to the very vanguard of study in staff idea.
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Extra resources for The subgroup structure of the finite classical groups
Sample text
Then XPI and XP2 are conJugate. (ii) Any G-invariant subspace of V is a sum of some of tbe Vi, Proof. (i) We use induction on t, and we define V;- 55 Basic Properties of the Classical Groups as G-modules. Proof. Assume that (Xp1)g = XP2 for some 9 E GL(V, F). Then there is a map B : X --+ X which satisfies (xpJ)g = (X())P2. Clearly () is a well-defined automorphism of X since thepi are faithful. Thus PI is equivalent to 8P2, so PI and P2 are quasiequivalent. The converse is just as easy, and the details are left to the reader.
2. But then KIT = G, and it now follows that I}(I = IG : ITIII< n ITI = IHI, a contradiction. 1 we have Hn < KnIT < IT. A-F. lQll write J('f! for the IT-associate \)\) ...... , of ]{. H. ii. If]{ is of type GL](2) I Sn in Cz(GL n(2)) with n ... " ...... ,,~ ............ " " ..... '"'' v j ..... " ..... J... 11,L,I,j, '-' /J tJ ot(q)@O~n(q). inIT with HO,i E C4 (G). :i of tensor even, product subgroups of type Sp2(q) ® SPn/2(Q) in C4 (IT). Furthermore, the last row of then ]{ nIT = Kfl x 2 ~ Sn x 2.
He Hi. H now columns 2 and 4 we give Cj and Cj, where Ho E Cj(IT) and ](0 E Cj(IT). Conditions' that ](1,](2,]{3, ](,1 are representatives for the four classes. In fact, 1':1 is Lhe under which the triple occurs appear in Column 6. Each row in the table corr Jji;';:: llnique member of [K l ] which contains HI; a similar statement holds for the other K j . to a unique triple (Ho,I{o,IT) up to conjugacy in IT, except when the term '(two)' 1::; i ::; 4, let ](j = KO,i n IT with ](O,i E C4 (G). '(four)' appears.