Symmetry And Group

Non-Commutative Harmonic Analysis and Lie Groups by J. Carmona, M. Vergne

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By J. Carmona, M. Vergne

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TERMINAL AND p-TERMINAL p-COMPONENTS 23 Furthermore, we see that I is in fact a p-component of Lp (CX (x)) ∩ CX (y). In particular, if K is quasisimple, then clearly I = K. 1(ii) applies to I. In this case we write Ky for the pumpup of I in CX (y), and sometimes call Ky a pumpup of K rather than of I. In particular, if Ky is a trivial pumpup of I, it follows that I maps onto both K/Op (K) and Ky /Op (Ky ), so that K/Op (K) ∼ = I/Op (I) ∼ = Ky /Op (Ky ). Note also that if a ∈ C(K, x) centralizes y, then a leaves I invariant and as I covers K/Op (K), a centralizes I/Op (I).

14 The “revised” quasithin problem is currently being investigated by the amalgam method15 . Stellmacher and Delgado have made inroads; but considerable parts of the problem remain to be done. Nevertheless, as noted in the Introduction, we are reasonably confident that this approach will succeed, and if necessary, one could 14 The rather technical restriction on the size of a Sylow 2-subgroup of N in the definition of σ(G) is chiefly for the sake of facilitating the amalgam method in the “revised” quasithin case.

It is the latter fact that explains the procedure for identifying the abstract simple group as a given group G(q) of Lie type. 2). [In practice, the analysis leads to the construction of a subgroup ˆα | α ∈ Σ G0 = X ˆ of G isomorphic to the target group G(q) or to a homomorphic image of G(q). There remains the entirely separate problem of proving that G = G0 . This remark applies equally well to the case in which the target group is an alternating group. ] ˆ α ) satisfy many conditions that are consequences The Xα (and likewise the X of the Steinberg relations.

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