all2 - Math 5125 Monday, October 3 Monday, October 3...

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Unformatted text preview: Math 5125 Monday, October 3 Monday, October 3 Exercise 6.3.7 on page 220 Prove that the following is a presentation for the quaternion group of order 8: Q 8 = a , b | a 2 = b 2 , a- 1 ba = b- 1 . Q 8 is a group with 8 elements, namely { 1 ,- 1 , ± i , ± j , ± k } with identity 1, center { 1 ,- 1 } , i 2 = j 2 = k 2 =- 1, i j = k , jk = i , ki = j . Let F denote the free group on { a , b } . Let us map F to Q 8 by sending a to i and b to j . Since Q 8 is generated by { i , j } , we obtain an epimorphism θ : F → Q 8 such that θ a = i and θ b = j . Let K = ker θ . Now θ ( a 2 ) = i 2 =- 1 = j 2 = θ ( b 2 ) and θ ( a- 1 ba ) = i- 1 ji =- i ji =- ki =- j = j- 1 = θ ( b- 1 ) , so if R is the normal subgroup generated by a 2 b- 2 , a- 1 bab , we see that R ⊆ K . Since F / K ∼ = Q 8 , we see that | F / K | = 8, so we will be done if we can show that | F / R | < 16 (because by Lagrange | F / R | has to be a multiple of | F / K | = 8, and | F / R | = 8 if and only if R = K ). Since a- 1 ba = b- 1 mod R , we see that a- 1 b 2 a = b- 2 mod R and hence a- 1 a 2 a = b- 2 = a- 2 mod R . We deduce that a 4 = b 4 = 1 mod R . Using ba = ab- 1 , we now see that every element of F can be written in the form a i b j mod R where 0 ≤ i , j ≤ 3. But a 2 b 2 = 1 mod R , consequently | F / R | ≤ 15. Therefore K is equal to the normal subgroup of F generated by { a 2 b- 2 , a- 1 bab } , and the result follows. Exercise 7.3.7 on page 248 Let R = { a b c d | a , b , d ∈ Z } be the subring of M 2 ( Z ) of upper triangular matrices. Prove that the map φ : R → Z × Z defined by φ : a b d → ( a , d ) is a surjective homomorphism and describe its kernel. Let x 1 = a 1 b 1 c 1 , x 2 = a 2 b 2 c 2 . Then φ ( x 1 + x 2 ) = ( a 1 + a 2 , d 1 + d 2 ) and φ ( x 1 )+ φ ( x 2 ) = ( a 1 , d 1 )+( a 2 , d 2 ) , so φ ( x 1 + x 2 ) = φ ( x 1 ) + φ ( x 2 ) which shows that φ respects addition. Also φ ( x 1 x 2 ) = ( a 1 a 2 , d 1 , d 2 ) and φ ( x 1 ) φ ( x 2 ) = ( a 1 , d 1 )( a 2 , d 2 ) , so φ ( x 1 x 2 ) = φ ( x 1 ) φ ( x 2 ) and thus φ also respects multipli- cation. This establishes that φ is a ring homomorphism. It is clearly onto, and its kernel consists of matrices of the form b where b is an arbitrary element of Z . Exercise 7.3.34 on page 250 Let I and J be ideals of the ring R with a 1. (a) Prove that I + J is the smallest ideal of R containing both I and J . (b) Prove that IJ is an ideal contained in I ∩ J . (c) Give an example where IJ = I ∩ J . (d) Prove that if R is commutative and if I + J = R , then IJ = I ∩ J . (a) Obviously I + J is an ideal containing both I and J . Suppose K is an ideal containing both I and J . Since K is closed under addition, it must contain I + J . Therefore I + J is the smallest ideal containing I and J ....
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This note was uploaded on 01/02/2012 for the course MATH 5125 taught by Professor Palinnell during the Fall '07 term at Virginia Tech.

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all2 - Math 5125 Monday, October 3 Monday, October 3...

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