Adv Alegbra HW Solutions 58

Adv Alegbra HW Solutions 58 - 58 (ii) Find two...

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58 (ii) Find two nonisomorphic groups of order 6. Solution. A cyclic group of order 6 is not congruent to S 3 , for the former is abelian and S 3 is not. 2.71 Prove that a dihedral group of order 4 is isomorphic to V , the 4-group, and a dihedral group of order 6 is isomorphic to S 3 . Solution. Absent. 2.72 Prove that any two dihedral groups of order 2 n are isomorphic. Solution. Absent. 2.73 De f ne a function f : S n GL ( n , R ) by f : σ 7→ P σ , where P σ is the matrix obtained from the n × n identity matrix I by permuting its columns by σ (the matrix P σ is called a permutation matrix ). Prove that f is an isomorphism from S n to a subgroup of GL ( n , R ) . Solution. Absent. 2.74 (i) Find a subgroup H S 4 with H = V but with H ±= V . Solution. An example is H ={ 1 ,( 12 ), ( 34 ), ( 1234 ) } . (ii) Prove that the subgroup H in part (i) is not a normal subgroup. Solution.
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This note was uploaded on 12/21/2011 for the course MAS 4301 taught by Professor Keithcornell during the Fall '11 term at UNF.

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