{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# ahw6 - Math 3124 Thursday October 20 Sixth Homework...

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

Math 3124 Thursday, October 20 Sixth Homework Solutions 1. Problem 17.9. Assume that G is a group with a subgroup H such that | H | = 6, [ G : H ] > 4, and | G | < 50. What are the possibilities for | G | ? By Lagrange’s theorem and using the given hypotheses, | G | = 6 [ G : H ] < 50 and [ G : H ] > 4. Therefore the possibilities for [ G : H ] are 5,6,7,8 and hence the possibilities for | G | are 30, 36, 42 and 48. 2. Problem 17.17. Find all the subgroups of Z 36 . Also construct the subgroup lattice. Since Z 36 is a cyclic group of order 36, there will be exactly one subgroup for each positive integer dividing 36. The integers dividing 36 are 2 a 3 b where a , b = 0 , 1 , 2, so there will be 3 * 3 = 9 subgroups. The subgroup lattice will look like [ 0 ] [ 12 ] [ 18 ] [ 6 ] [ 2 ] [ 9 ] [ 4 ] [ 3 ] [ 1 ] 3. From ae = c , we see that neither a nor e is the identity, and from bd = c , we see that neither b nor d is the identity. Therefore c is the identity. Since c is the identity, ae = c yields ea = c , and bd = c yields db = c . We now have the following partial Cayley table a b c d e a a c b b c c a b c d e d c d e c e There are now two ways to complete the table. The first possibility is that

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