hw08 - ideal in Z (c.f. section I.3) compare with the...

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Modern Algebra 1 Homework 4.1 Spring 2010 Due February 10 Exercise 1. Recall the group Q = h A,B i where A = ± 0 1 - 1 0 ² and B = ± 0 i i 0 ² (this group is known as the quaternion group , by the way). Find all the subgroups of Q and draw the subgroup lattice for Q . [ Note: It may be useful to recall that Q = I, ± A, ± B, ± AB } , where I is the 2 × 2 identity matrix.] Exercise 2. Determine all of the subgroups of Z that contain h 90 i and draw a lattice of these subgroups. This lattice is actually “isomorphic” to the subgroup lattice of Z 90 , as we’ll see later. Exercise 3. How does Lang’s definition of an
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Unformatted text preview: ideal in Z (c.f. section I.3) compare with the notion of a subgroup of Z ? How does his Theorem 3.1 compare to the theorem on the subgroups of Z that we proved in class? Exercise 4. Show that Q (with addition) is not nitely generated , i.e. given any nite set r 1 ,r 2 ,...,r n Q then h r 1 ,r 2 ,...,r n i 6 = Q . [ Suggestion: Show that there are only a limited number of denominators that can be obtained from integral linear combinations of r 1 ,r 2 ,...,r n .]...
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This note was uploaded on 02/29/2012 for the course MATH 3362 taught by Professor Ryandaileda during the Spring '10 term at Trinity University.

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