By Clara Löh

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**Example text**

N}, j = k} is “the” complete graph on n vertices. If n > 0, then Kn has exactly n vertices (each of degree n − 1) and 1/2 · n · (n − 1) edges. 1. : Some graphs Kn,m := {(1, 0), . . , (n, 0), (1, 1), . . , (m, 1)}, {{(j, 0), (k, 1)} | j ∈ {1, . . , n}, k ∈ {1, . . , m}} is “the” complete bipartite graph graph. If n, m > 0, then Kn,m has exactly n + m vertices and 1/2 · n · m edges. 5 (Graph isomorphisms). Let X = (V, E) and X = (V , E ) be graphs. 1 The problem to decide whether two given graphs are isomorphic or not is a difficult problem – in the case of finite graphs, this problem seems to be a problem of high algorithmic complexity, though its exact complexity class is still unknown [68].

2. For which n ∈ N is the group Gn Abelian? 25 (Braid groups**). For n ∈ N the braid group on n strands is defined by Bn := s1 , . . , sn−1 {sj sj+1 sj = sj+1 sj sj+1 | j ∈ {1, . . , n − 2}} {sj sk = sk sj | j, k ∈ {1, . . , n − 1}, |j − k| ≥ 2} . 1. Show that Bn −→ Z sj −→ 1 defines a well-defined group homomorphism. For which n ∈ N is this homomorphism surjective? 2. Show that Bn −→ Sn sj −→ (j, j + 1) 44 2. Generating groups defines a well-defined surjective homomorphism onto the symmetric group Sn .

1 Products and extensions The simplest type of group constructions are direct products and their twisted variants, semi-direct products. 32 2. 1 (Direct product). Let I be a set, and let (Gi )i∈I be a family of groups. The (direct) product group i∈I Gi of (Gi )i∈I is the group whose underlying set is the cartesian product i∈I Gi and whose composition is given by pointwise composition: Gi × i∈I Gi −→ i∈I Gi i∈I (gi )i∈I , (hi )i∈I −→ (gi · hi )i∈I . , homomorphisms to the direct product group are in one-to-one correspondence to families of homomorphisms to all factors.