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Unformatted text preview: Math 5125 Monday, August 22 August 22, Ungraded Homework Exercise 3.1.36 on page 89 Prove that if G / Z ( G ) is cyclic, then G is abelian. Write Z = Z ( G ) . If G / Z is cyclic, then G / Z = xZ for some x G . This means that the left cosets of Z in G are of the form x a Z for some a Z . Therefore every element of G is of the form x a z for some z Z . Suppose we have another such element, say x b w . Then ( x a z )( x b w ) = zwx a x b = wzx b x a = ( wx b )( zx a ) , consequently any two elements of G commute and we have proven that G is abelian. Exercise 3.2.4 on page 95 Show that if G = pq for some primes p and q (not necessarily distinct), then either G is abelian or Z ( G ) = 1. By Lagranges theorem, | Z ( G ) | = 1 , p , q or pq . If | Z ( G ) | = pq , then G is abelian, so we may assume without loss of generality that | Z ( G ) | = p . But then | G / Z ( G ) | = q . Since groups of prime order are cyclic, we deduce from the previous exercise that G is abelian and the result follows. Let G be a finite group, let H G and let K G . Prove that | HK : H | = | K : K H | . Note that HK and K H are subgroups of G (the former by, for example, the second iso- morphism theorem), so the result to be proven makes sense. By the second isomorphism theorem, HK / K = H / H K . Since everything in sight is finite, we see that | HK | / | K | = | H | / | H K | and hence | HK | / | H | = | K | / | K H | . The result follows. Math 5125 Wednesday, August 24 August 24, Ungraded Homework Exercise 3.3.3 on page 101 Prove that if H is a normal subgroup of G of prime index p , then for all K G either (i) K H or (ii) G = HK and | K : K H | = p . We know that HK G , because H , K G and one of them is normal. Obviously HK H . Also | G / H | = p and p is prime, so by Lagranges theorem G / H has only two subgroups, namely H / H and G / H . By subgroup correspondence theorem, we now see that HK = H or G . If K is not contained in H , then we cannot have HK = H and we deduce that HK = G . By the second isomorphism theorem we have HK / H = K / K H , consequently | G / H | = | K / K H | and the result follows. Let H G be groups such that G / H = Z / 3 Z Z / 3 Z . Prove that G has at least four normal subgroups of index 3. Let H G such that G / H = Z 3 Z 3 . We need to prove that G has at least 4 normal sub- groups of index 3. It is easily checked that Z 3 Z 3 has 4 subgroups of order 3. Since Z 3 Z 3 has order 9, these subgroups will all have index 9/3 = 3. Also all these subgroups are normal because Z 3 Z 3 is an abelian group. Thus G / H has 4 normal subgroups of index 3. It now follows from the subgroup correspondence theorem that G has 4 normal subgroups of index 3 containing H , which proves the result....
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