By Robert A. Conover

**Publish yr note:** initially released in 1975

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Students needs to end up all the theorems during this undergraduate-level textual content, which positive aspects huge outlines to aid in research and comprehension. Thorough and well-written, the remedy presents enough fabric for a one-year undergraduate path. The logical presentation anticipates students' questions, and entire definitions and expositions of themes relate new innovations to formerly mentioned subjects.

Most of the fabric specializes in point-set topology apart from the final bankruptcy. themes comprise units and features, countless units and transfinite numbers, topological areas and uncomplicated innovations, product areas, connectivity, and compactness. extra topics contain separation axioms, entire areas, and homotopy and the basic workforce. a number of tricks and figures remove darkness from the text.

**Read Online or Download A First Course in Topology: An Introduction to Mathematical Thinking PDF**

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**Extra info for A First Course in Topology: An Introduction to Mathematical Thinking**

**Example text**

14. C HARACTERS OF SYMMETRIC GROUPS . In this section we will assume that k = C, the complex numbers. Let G = Sn be the symmetric group on n letters. The conjugacy classes of an element of σ ∈ G are given by the cycle type of σ, which (if we arrange the cycles of σ in decreasing order of size) just gives a partition of n, that is, a sequence (λ1 , λ2 , . . , λr ) of positive integers, such that λ1 ≥ λ2 ≥ . . ≥ λr , where r i=1 λi = n. Thus for example, representatives for the conjugacy classes in S4 are e, (12), (12)(34), (123), (1234), corresponding to the partitions (1, 1, 1, 1), (2, 1, 1), (2, 2), (3, 1), (4) of 4.

These are the only one-dimensional representations of Sn (up to isomorphism) since if ψ : Sn → GL1 (k) = k× which is Abelian, and so ker(ψ) contains all commutators {[g1 , g2 ] = g1 g2 g1−1 g2−1 : g1 , g2 ∈ Sn }, and hence the subgroup that they generate, the commutator or derived subgroup. For Sn this is the whole alternating group Altn , hence ψ comes from a representation of Sn /An ∼ = Z/2Z, and these are the trivial and sign representation respectively. For n = 3 these are all the irreducible representations (there are only three conjugacy classes in this case).

The character of the regular representation is easy to compute. Indeed using the basis GROUP REPRESENTATION THEORY, IMPERIAL ’09. 37 {eg : g ∈ G} we see that tr(g, k[G]) is zero unless g = e (since the matrix corresponding to g in this basis will consist of 1s and 0s with a 1 in the diagonal exactly when gh = h, that is when g = e). Thus |G|, if g = e, 0, otherwise. χk[G] (g) = It follows from this, and the fact that χV (e) = dim(V ) for every representation V , that dim(HomG (k[G], V )) = χk[G] , χV = dim(V ).