By Tammo tom Dieck

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

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**Example text**

This would give 12 = X A = 0 which is impossible. Thus, if ad - be = 0 then the matrix A does not have an inverse. B = /2B = (XA)B The number ad - be is clearly significant here and is known as the determinant of A and written det A. The idea of a determinant generalizes to n x n matrices. The general definition and properties can be found in the book in this series by Allenby. • Theorem I Let A, B E Mn(l)~). Then (i) A is invertible if and only if det A =j:. O. (ii) det A det B = det(AB). (iii) If A and B are invertible matrices then AB is invertible.

Then h = 2n for some n 2n for some m, n E E 7L and h- 1 = 2- n 7L and E H. Chapter 5 Subgroups Thus, by the subgroup criterion, H is a subgroup of {IR\{O}, x). This subgroup, built from the single element 2, is an example of what we shall later call a cyclic subgroup. Roots of unity = Fix a positive integer n and, in C\{O} under multiplication, let H {z E C : zn = I}. Thus, H is the set of all complex nth roots of unity. For example, if n = 4 then H = {I, i, -1, -i}. We check SGl,SG2,SG3 as follows..

SGl: 0 = m x SG2: Let a, b SG3: Let a ° m71.. E E n7L; a so m7L ::j:. 0. = mn, b = mk for some n, k E 7l... Then a + b = m(n + k) E m7L. = mn E m7L. Then -a = m(-n) E m71... By the subgroup criterion, m71.. is a subgroup of 2. In the general linear group G L n (~), let H denote the set of all n x n real matrices with determinant 1, that is, H = {A E GL n (lR) : det A = I}. by Theorem l(i) of Chapter 3, H is a subset of G L n (IR). SGl: H =f. 0 because In E H. SG2: Let A, B E H, that is det A Hence, AB E H.