ProblemSet8

# ProblemSet8 - p-subgroups of G(h If G and H are ﬁnite...

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Homework 8 Due: Friday, April 2, 2010 1. Answer True or False and justify your answer in a sentence or two. (a) If G is a ﬁnite group and p is prime then G has exactly one Sylow p -subgroup. (b) If G is ﬁnite abelian group and p is prime then G has exactly one Sylow p -subgroup. (c) If G is a ﬁnite group and p is prime then G has at least one Sylow p -subgroup. (d) If a group G acts on a set X and if x,y X belong to the same orbit then G x and G y are conjugate subgroups of G . (e) If H G then N G ( H ) C G . (f) If H G then H C N G ( H ). (g) A Sylow p -subgroup of G contains all the other
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Unformatted text preview: p-subgroups of G . (h) If G and H are ﬁnite groups of the same order, then, for every prime p , their Sylow p-subgroups have the same order. (i) There is a group of order 400 having exactly 8 Sylow 5-subgroups. [9 pts] 2. Give an example of a ﬁnite group G having 3 Sylow p-subgroups (for some prime p ) P , Q and R such that P ∩ Q = { e } and P ∩ R 6 = { e } . [3 pts] 3. Prove that there are no simple groups of order 300, 312, 616 or 1000. [3 pts]...
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