hw11 - Kra 5 Let GL(2 Z 2 represent the group of 2 × 2...

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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 coeFcients 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. ±or every divisor of | Z 24 | = 24, identify a subgroup of Z 24 with that cardinality. (This should be easy). 4. ±or 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
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Unformatted text preview: Kra.) 5. Let 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? 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...
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