aug24 - Math 5125 Wednesday August 24 August 24 Ungraded...

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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 Lagrange’s 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
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