A course in abstract algebra by Nicholas Jackson

By Nicholas Jackson

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49 The tetrahedron ∆3 has twelve direct isometries: • The identity isometry e • Eight rotations ri± through angles of ± 2π 3 around one of four axes (i = 1, . . , 4) passing through a vertex and the centre of the opposite face • Three rotations s A , s B and sC through an angle of π around one of three axes passing through the midpoints of opposite edges r1± The multiplication table for this group Isom+ (∆3 ) is as follows: e r1+ r1− r2+ r2− r3+ r3− r4+ r4− sA sB sC e e r1+ r1− r2+ r2− r3+ r3− r4+ r4− sA sB sC r1+ r1+ r1− e r3− sC r4− sB r2− sA r3+ r2+ r4+ r1− r1− e r1+ sB r4+ sA r2+ sC r3+ r4− r3− r2− r2+ r2+ r4− sC r2− e r1− sA r3− sB r4+ r1+ r3+ r2− r2− sB r3+ e r2+ sC r4+ sA r1+ r3− r4− r1− r3+ r3+ r2− sB r4− sA r3− e r1− sC r1+ r4+ r2+ r3− r3− sA r4+ sC r1+ e r3+ sB r2+ r2− r1− r4− r4+ r4+ r3− sA r1− sB r2− sC r4− e r2+ r3+ r1+ r4− r4− sC r2+ sA r3+ sB r1+ e r4+ r1− r2− r3− sA sA r4+ r3− r3+ r4− r2+ r1− r1+ r2− e sC sB sB sB r3+ r2− r4+ r1− r1+ r4− r2+ r3− sC e sA sC sC r2+ r4− r1+ r3− r4+ r2− r3+ r1− sB sA e r3± r2± r4± sB sA sC In this table, the operation in row a and column b is the product ab; that is, the isometry a followed by the isometry b.

What happens to the rest of the Stifle is an anagram of itself. — Anonymous triangle is determined completely by where the vertices go. So we could describe the elements of R3 , the rotation transformations on the equilateral triangle, solely as permutations of the set {0, 1, 2}. Similarly, the dihedral group D3 can also be regarded as permutations of the set {0, 1, 2}. The identity transformation leaves all three points unchanged, the clockwise and anticlockwise rotations permute them cyclically, and each of the three reflections leave one point unchanged and swap the other two points round.

This has order–n rotational symmetry, and n axes of symmetry. In the case where n is odd, each of these axes passes through the midpoint of one side and the opposite vertex. 5: The multiplication table for the dihedral group D3 through pairs of opposite vertices, and the other half pass through the m2 midpoints of opposite sides. 43 The dihedral group Dn = {e, r, . . , r n−1 , m1 , . . , mn } is the symmetry group of the regular plane n–gon. Here, e is the identity, r represents an anticlockwise rotation through an angle of 2π n , and mk denotes a reflection in the line which makes an angle of kπ n with the horizontal.

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