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[solution] » This about abstract algebra. Reference book: Contemporary Abstract Algebra of Joseph A.Gallian. <


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This about abstract algebra. Reference book: Contemporary Abstract Algebra of Joseph A.Gallian. 


(a) Let A be any counter-symmetric deranged matrix. Write A as a linear combination of the

 

matrices I, P(123) and P(132)

 

(b) Let A be any symmetric deranged matrix. Write A as a linear combination of the matrices

 

P(12) , P(23) , P(13) .

 

(c) Let D be the subgroup of all deranged matrices in GL(3, R); let D1 be the subgroup of

 

counter-symmetric deranged matrices in D. Show that for any odd permutation ? ? S3 ,

 

(i)

 

3. Let D = D1 ? (P? D1 ) (ii) D1 C D (iii) D/D1 ? Z2 a c b

 

a b c

 

A = b a c or b c a c b a

 

c a b be a deranged matrix.

 

(a) Show that A is orthogonal if and only if a2 + b2 + c2 = 1 and a + b + c = ±1.

 

(b) Find three distinct2 triples (a, b, c) consisting of rational numbers that satisfy the conditions

 

of part (a).

 

4. As demonstrated in class the symmetry group of a cube, S(C), consists of the following 48 matrices 0

 

0 ±1

 

0 ±1 0

 

±1 0

 

0 0 ±1 0 , 0

 

0 ,

 

0 ±1 , ±1 0

 

0 ±1 0

 

±1 0

 

0

 

0

 

0 ±1 ±1 0

 

0

 

0

 

0 ±1

 

0 ±1

 

0 0

 

0 ±1 , 0 ±1 0 , ±1

 

0

 

0 ,

 

0 ±1 0

 

±1 0

 

0

 

0

 

0 ±1

 

with usual matrix multiplication. Show that

 

S(C) ? S4 ? Z2 .

 

Recall: the rotations in S(C) are the 24 matrices with determinant equal to 1. 2 That is, (1, 0, 0) is one solution, but (0, ±1, 0) counts as the same solution.

 


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