{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

hw11_2 - GL(2 Z 2 represent the group of 2 × 2 matrices...

This preview shows page 1. Sign up to view the full content.

MAT 312/AMS 351 – Fall 2010 Homework 11 1. A linear fractional transformation f ( x ) is a function of the form f ( x ) = ax + b cx + d where a,b,c,d, are real numbers satisfying ad bc = 1. Show that if f ( x ) as above and g ( x ) = ( ex + f ) / ( gx + h ) are linear fractional transformations, so is their composition f g (where f g ( x ) = f ( g ( x )) as usual). Hint: the coefficients in f g are related to those in f and g by matrix multiplication, where the matrices have determinant 1. 2. Show that in Z * 13 the elements { 1 , 3 , 4 , 9 , 10 , 12 } form a subgroup. Call it H . Show that H = < 4 > , i.e. that the elements 4 0 = 1 , 4 1 = 4 , 4 2 = 3 , etc. make up all of H . Conclude that H is isomorphic to the additive group Z 6 . Explain carefully. 3. For every divisor of | Z 24 | = 24, identify a subgroup of Z 24 with that cardinality. (This should be easy). 4. For every divisor of | S (4) | = 24, identify a subgroup of | S (4) | with that cardinality. (Not so easy. Hints can be found on pages 82 and 92 in Kra.) 5. Let GL (2 , Z 2 ) represent the group of 2 × 2 matrices with coefficients
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: GL (2 , Z 2 ) represent the group of 2 × 2 matrices with coeFcients in Z 2 and determinant 1 (calculated mod 2). What is the cardinality n of this group? List its elements. Write out its multiplication table. Identify the group with a subgroup of S ( n ) (as usual, thinking of each element as de²ning, by left-multiplication, a permutation of the ele-ments of the group). Is this a group we have seen before, perhaps in di³erent clothing? Hint: consider the action by left-multiplication of GL (2 , Z 2 ) on the column vectors 1 = p 1 P , 2 = p 1 P , 3 = p 1 1 P . 6. In the permutation group S (4), let H represent the subgroup H = { e, (1234) , (13)(24) , (1432) } . H should have 6 left cosets. What are they? Describe them by listing their elements. 1...
View Full 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