Adv Alegbra HW Solutions 82

Adv Alegbra HW Solutions 82 - 82 a subgroup of S6 , for...

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82 a subgroup of S 6 , for each ϕ 6(π 6 ) is a permutation of the vertices (draw a hexagon similar to Figure 2.15). There are 12 elements: 6 rotations: ( 1 ), (v 0 v 1 v 2 v 3 v 4 v 5 ), (v 0 v 2 v 4 )(v 1 v 3 v 5 ), (v 0 v 3 )(v 1 v 4 )(v 2 v 5 ), (v 0 v 4 v 2 )(v 1 v 5 v 3 ), (v 0 v 5 v 4 v 3 v 2 v 1 ) ; 6re f ections: (v 1 v 5 )(v 2 v 4 ), (v 0 v 2 )(v 3 v 5 ), (v 0 v 4 )(v 1 v 3 ), (v 0 v 1 )(v 2 v 5 )(v 3 v 4 ), (v 0 v 3 )(v 1 v 2 )(v 4 v 5 ), (v 0 v 5 )(v 1 v 4 )(v 2 v 3 ). (iii) How many bracelets are there with exactly 6 beads having 1 red bead, 2 white beads, and 3 blue beads? Solution. Absent. Exercises for Chapter 3 3.1 True or false with reasons. (i) The subset { r + s π : r , s Q } is a subring of R . Solution. False. (ii) Every subring of a domain is a domain. Solution. True. (iii) The zero ring is a subring of
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