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Unformatted text preview: 5.6. ADDITIONAL EXERCISES FOR CHAPTER 5 263 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 p k for some k . Show that
H Â P.
5.6.18. Show that a group of order 2n 5m , m; n
subgroup. Can you generalize this statement? 1, has a normal 5–Sylow 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
gP g 1 .
g 2G ...
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