By Tammo tom Dieck

ISBN-10: 3540505288

ISBN-13: 9783540505280

This ebook is a jewel– it explains vital, worthwhile and deep themes in Algebraic Topology that you just won`t locate somewhere else, conscientiously and in detail."""" Prof. Günter M. Ziegler, TU Berlin

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**Additional resources for Algebraic Topology and Transformation Groups**

**Example text**

2 (cf. 14). r-local structure for r = p. Next we consider the case when the prime r is diﬀerent from the natural characteristic p of G. Recall that the r-elements of G are semisimple, that is, diagonalizable over the algebraic closure F of F . For expository purposes, consider our standard example G = GL(V ), where we have g ∈ G of order r, and diagonalizable even over F . Then the r-th roots of 1 {λi : 1 ≤ i ≤ r} are the (potential) eigenvalues of g, and V = V1 ⊕ · · · ⊕ Vr , where Vi is the g-eigenspace corresponding to λi .

It may not be immediately evident how the actual proof that developed ﬁts into our six-step strategy above, so at relevant points in our narrative, we will discuss how the actual proof required some adjustments to compensate for complications to our idealized strategy. As a brief preview, we provide below a quick informal summary of the main stages; in particular we supply some names and reference numbers for the major subtheorems which will emerge during the course of the subsequent discussion.

In doing so, we are emphasizing that these results • are not directly concerned with groups of GW type; and • are used in important ways in both the Odd and Even Cases of the CFSG. 1: The previous section treated the 2-rank 0 subcase of the Small Odd Case (2-rank at most 2) in Step 2 of the Strategy. Before completing that Step in the following section, we ﬁrst discuss uniqueness subgroups—corresponding to the exceptional branch in Step 1 of the Strategy. The Strongly Embedded Theorem as the main 2-Uniqueness Theorem.

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