5.6. ADDITIONAL EXERCISES FOR CHAPTER 5
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5.6. ADDITIONAL EXERCISES FOR CHAPTER 5
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(b)
A5
(c) h.1 2 3 4 5/; .2 3 5 4/i Š Z4 Ë Z5 (and its five conjugates) (d) h.1 2 3 4 5/; .2 5/.3 4/i Š D5 (and its five conjugates) (e) h.1 2 3 4 5/i Š Z5 (and its five conjugates) Remark 5.5.4. There are 16 conjugacy classes of transitive subgroups of S6, and seven of S7. (See J. D. Dixon and B. Mortimer, Permutation Groups, Springer–Verlag, 1996, pp. 58–64.) Transitive subgroups of Sn 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 fgHg 1 W g 2 Gg. As usual, G=H denotes the set of left cosets of H in G, G=H D fgH W g 2 Gg.
#.G=H / (a) Show that D ŒNG.H / W H .
#Y (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 NG.H /. Show that, in general, the map is ŒNG.H / W H to one (i.e., the preimage of each element of Y has size ŒNG.H / W H ).
Definition 5.6.1. Suppose a group G acts on sets X and Y . We say that a map ' W X ! Y is G-equivariant if for all x 2 X, '.g x/ D g .'.x//: 5.6.2. Let G act transitively on a set X . Fix x0 2 X, let H D Stab.x0/, and let Y denote the set of conjugates of H in G. Show that there is a G-equivariant surjective map from X to Y given by x 7! Stab.x/, and this map is ŒNG.H / W H to one.
5.6.3. Let D4 S4 be the subgroup generated by .1234/ and .14/.23/.
Show that NS .D
4
4/ D D4. Conclude that there is an S4-equivariant bi jection from S4=D4 onto the set of conjugates of D4 in S4.
5.6.4. Let G be the rotation group of the tetrahedron, acting on the set of faces of the tetrahedron. Show that map F 7! Stab.F / is bijective, from the set of faces to the set of stabilizer subgroups of faces.
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5. ACTIONS OF GROUPS
5.6.5. Let G be the rotation group of the cube, acting on the set of faces of the cube. Show that map F 7! Stab.F / is 2-to-1, from the set of faces to the set of stabilizer subgroups of faces.
5.6.6. Let G D Sn and H D Stab.n/ Š Sn 1. Show that H is its own normalizer, so that the cosets of H correspond 1-to-1 with conjugates of H . Describe the conjugates of H explicitly.
5.6.7. Identify the group G of rotations of the cube with S4, via the action on the diagonals of the cube. G also acts transitively on the set of set of three 4–fold rotation axes of the cube; this gives a homomorphism of S4 into S3.
(a) Compute the resulting homomorphism of S4 to S3 explicitly.
(For example, compute the image of a set of generators of S4.)
Show that is surjective. Find the kernel of .
(b) Show that the stabilizer of each 4–fold rotation axis is conjugate to D4 S4.
(c) Show that L 7! Stab.L/ is a bijection between the set of 4–fold rotation axes and the stabilizer subgroups of these axes in G.
This map is G-equivariant, where G acts on the set of stabilizer subgroups by conjugation.
5.6.8. Let H be a proper subgroup of a finite group G. Show that G contains an element that is not in any conjugate of H .
5.6.9. Find all (2- and 3-) Sylow subgroups of S4.
5.6.10. Find all (2- and 3-) Sylow subgroups of A4.
5.6.11. Find all (2- and 3-) Sylow subgroups of D6.
5.6.12. Let G be a finite group, p a prime, P a p–Sylow subgroup of G, and N a normal subgroup of G. Show that PN=N is a p–Sylow subgroup of G=N and that P \ N is a p–Sylow subgroup of N .
5.6.13. Let G be a finite group, p a prime, and P a p–Sylow subgroup of G. Show that NG.NG.P // D NG.P /.
Let p be a prime. Recall that a group (not necessarily finite) is called a p-group if every element has finite order pk for some k 0.
5.6.14. Show that a finite group G is a p-group if, and only if, the order of G is equal to a power of p.
5.6.15. Let N be a normal subgroup of a group G (not necessarily finite).
Show that G is a p-group if, and only if, both N and G=N are p-groups.
5.6.16. Let H be a subgroup of a finite p-group G, with H ¤ feg. Show that H \ Z.G/ ¤ feg.
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5.6.17. Let G be a finite group, p a prime, and P a p–Sylow subgroup.
Suppose H is a normal subgroup of G of order pk for some k. Show that H P .
5.6.18. Show that a group of order 2n5m, m; n 1, has a normal 5–Sylow subgroup. Can you generalize this statement?
5.6.19. Show that a group G of order 56 has a normal Sylow subgroup.
Hint: Let P be a 7–Sylow subgroup. If P is not normal, count the elements
[
in gP g 1.
g2G
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