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

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern