By American Mathematical Society
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Additional resources for A Crash Course on Kleinian Groups
How would the other three semi-direct products be denoted in an unambiguous manner? (3) Let K = e 0 1 2 3 0 2 3 1 0 3 1 2 ≤ G S4 be the Klein fourgroup and let S3 A = 1 2 3 2 3 ≤ G. Show that K A G, where xa for x ∈ K a ∈ A denotes the conjugate of x by a. (4) Let K be a finite group and let A = Aut K be the automorphism group of K. The semi-direct product K A is called the holomorph of K. Let K 2n , an elementary abelian group of order 2n , then A = Aut K Ln 2 , the general linear group in n dimensions over the field of order 2.
In a moment, we shall see Moreover, a e x = a x and so K A = A 48 The (involutory) progenitor that the subgroup K is normal, and so, although the colon notation reverses the order of the semi-direct product in a slightly unfortunate manner, we ˆK ˆ Indeed, note that ˆ =K ˆ A. have A a x −1 a x = a−1 x−1 y a−1 ˆ x−1 ya x ∈ K a ya x = ˆ and so K K A . So, K A is a product of a normal subgroup isomorphic to K and a subgroup isomorphic to A which intersect trivially. In practice, we ˆ with A, and write the element a x simply ˆ with K and A shall identify K as ax.
4) the three edges are coloured red (6), black (0) and yellow (2), which is a line of the 7-point projective plane. Just as it was easy to read off generators for M12 from the vertex-colouring of the dodecahedron, we may now read off generators for M24 from the edge-colouring of the Klein map. Corresponding to each colour we define a permutation as follows. Choose a colour and interchange every pair of faces which are separated by an edge of that colour. 4. Alternative colouring of the Klein map.
A Crash Course on Kleinian Groups by American Mathematical Society