5.6. ADDITIONAL EXERCISES FOR CHAPTER 5 261 (c) h .1 2 3 4 5/;.2 3 5 4/ i Š Z 4 Ë Z 5 (and its ﬁve conjugates) (d) h .1 2 3 4 5/;.2 5/.3 4/ i Š D 5 (and its ﬁve conjugates) (e) h .1 2 3 4 5/ i Š Z 5 (and its ﬁve conjugates) Remark 5.5.4. There are 16 conjugacy classes of transitive subgroups of S 6 , and seven of S 7 . (See J. D. Dixon and B. Mortimer, Permutation Groups , Springer–Verlag, 1996, pp. 58–64.) Transitive subgroups of S n at least for n ± 11 have been classiﬁed. Consult Dixon and Mortimer for further details. 5.6. Additional Exercises for Chapter 5 5.6.1. Let G be a ﬁnite group and let H be a subgroup. Let Y denote the set of conjugates of H in G , Y D f gHg ± 1 W g 2 G g . As usual, G=H denotes the set of left cosets of H in G , G=H D f gH W g 2 G g . (a) Show that # .G=H/ # Y D ŒN G .H/ W HŁ . (b) Consider the map from G=H to Y deﬁned by gH 7! gHg ± 1 . Show that this map is well deﬁned and surjective. (c) Show that the map in part (b) is one to one if, and only if, H D N G .H/ . Show that, in general, the map is ŒN G .H/ W HŁ to one (i.e., the preimage of each element of
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