hw2.pdf - MODERN ALGEBRA 1 HOMEWORK 2 The problem numbers...

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MODERN ALGEBRA 1: HOMEWORK 2 The problem numbers refer to Artin’s Algebra (2nd edition). (1) Chapter 1: 1.7 (2) Chapter 1: 4.4 (3) Chapter 1: 6.2 ( Hint: Use cofactors – Theorem 1.6.9 ). (4) Chapter 2: 2.4 (5) Suppose ( ab ) 2 = a 2 b 2 for all a , b in a group G . Prove that G is abelian. (6) Prove that the set of 3 × 3 matrices of the form 1 a b 1 c 1 , a , b , c R is a subgroup of GL 3 ( R ) . It is called the Heisenberg group . (7) Recall that the Klein four group consists of the matrices ± 1 0 0 ± 1 . Find all the subgroups of the Klein four group. (8) Suppose G is a finite group whose order is even. Show that there exists an element
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