1 Let H(F) = { 0 1 X = 0 a 1 001 :) | a,...

 

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1 Let H(F) = { 0 1 X = 0 a 1 001 :) | a, b, c e F) called the Heisenberg group over F. Let b a b 1 d 1 C 0 1 f be elements of H(F). 0 0 1 00 1 (a) Compute the matrix product XY and deduce that H(F) is closed under matrix mul- tiplication. Exhibit explicit matrices such that XY YX (so that H(F) is always non-abelian). and Y = (b) Find an explicit formula for the matrix inverse X-1 and deduce that H(F) is closed under inverses. (c) Prove the associative law for H(F) and deduce that H(F) is a group of order |F1. (Do not assume that matrix multiplication is associative.) (d) Find the order of each element of the finite group H (Z/2Z). (e) Prove that every nonidentity element of the group H(R) has infinite order. 1 Let H(F) = { 0 1 X = 0 a 1 001 :) | a, b, c e F) called the Heisenberg group over F. Let b a b 1 d 1 C 0 1 f be elements of H(F). 0 0 1 00 1 (a) Compute the matrix product XY and deduce that H(F) is closed under matrix mul- tiplication. Exhibit explicit matrices such that XY YX (so that H(F) is always non-abelian). and Y = (b) Find an explicit formula for the matrix inverse X-1 and deduce that H(F) is closed under inverses. (c) Prove the associative law for H(F) and deduce that H(F) is a group of order |F1. (Do not assume that matrix multiplication is associative.) (d) Find the order of each element of the finite group H (Z/2Z). (e) Prove that every nonidentity element of the group H(R) has infinite order.

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a To compute the matrix product XY we perform the matrix multiplication XY 1 a b 0 1 c 0 0 1 1 d e 0 1 f 0 0 1 Multiplying the corresponding entries we get XY 1 ad a bd ac b ae bc c 0 1 c f 0 0 1 From ... View the full answer