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By Tammo tom Dieck
This ebook is a jewel– it explains very important, necessary and deep themes in Algebraic Topology that you simply won`t locate somewhere else, rigorously and in detail."""" Prof. Günter M. Ziegler, TU Berlin
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Extra resources for Algebraic Topology and Transformation Groups
Show that S forms a group. (ii) If T is a semigroup, and for all a ∈ T there is a unique a ∗ ∈ T satisfying aa ∗ a = a, prove that T is a group. 5 If H, J ≤ G and in (iii) p is a prime, show that (i) If H is a subset of J , then H ≤ J . (ii) H ∩ J = H if, and only if, H ≤ J . (iii) If o(H ) = o(J ) = p, then either H = J or H ∩ J = e . 6 Prove that if G is a group and S ≤ G, then SS = S. Conversely, if T is a non-empty finite subset of G and T T = T , prove that T ≤ G. Is this true if T is infinite?
24, each of these cosets has the same cardinality (number of elements), that is o(H ), the theorem follows. This result is particularly useful in the finite case where it shows that H can only be a subgroup of G if o(H ) | o(G); that is, the prime factorisation of the order of a group G is an important invariant of G. For instance, a group of order 30 cannot have subgroups of order 4, 7, 8, 9, 11, . . , 29. Also, it cannot have elements of order 4, 7, . . 19. In the infinite case, the theorem shows that either the order of the subgroup, or the index (or both), must be infinite.
See page 12 for the case n = 3. Note that Sn is non-Abelian if n > 2, and has order n! (count all possible maps). 11. 2 Examples 21 Examples from Analysis Some classes of functions form groups. For example, let Z denote the set of all continuous, strictly-increasing functions f which map [0, 1] onto [0, 1], and satisfy f (0) = 0 and f (1) = 1. This set Z forms a group if the operation is taken to be composition of functions (the identity function f0 , where f0 (x) = x for all x, acts as the neutral element, and inverses exist as the functions f are continuous and strictly monotonic).
Algebraic Topology and Transformation Groups by Tammo tom Dieck