HW2_Solutions

HW2_Solutions - Math 541 Solutions to HW#2 1 Let GL2(Z2...

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Math 541 Solutions to HW #2 1. Let GL 2 ( Z 2 ) denote the collection of 2 × 2 matrices with entries in Z 2 which have non-zero determi- nant.(We listed these matrices out in class.) (a) Make a multiplication table for GL 2 ( Z 2 ). Let A = ± 1 0 0 1 ² , B = ± 1 1 1 0 ² , C = ± 0 1 1 1 ² , D = ± 0 1 1 0 ² , E = ± 1 0 1 1 ² , and F = ± 1 1 0 1 ² . The multiplication table may be expressed as follows: A B C D E F A A B C D E F B B C A F D E C C A B E F D D D E F A B C E E F D C A B F F D E B C A (b) Which pairs of matrices satisfy a · b = b · a ? For all b GL 2 ( Z 2 ), A · b = b · A . That is, every element commutes with the identity. Every element also commutes with itself. The only other pair that commutes is ( B,C ). (c) Are there any elements which commute with every other matrix? That is, find all elements a in GL 2 ( Z 2 ) such that a · b = b · a for every b in GL 2 ( Z 2 ). The only element which commutes with every other element in this table is the identity. (d) For each matrix a , compute a , a 2 , a 3 , and so on until the pattern is clear. Determine the length of the repeating cycle for each matrix. Let a = A. Then we have A, A, A,. ... Cycle length is 1. Let a = B. Then we have B, C, A, B, C,. ... Cycle length is 3. Let a = C. Then we have C, B, A, C, B,. ... Cycle length is 3. Let a = D. Then we have D, A, D, A,. ... Cycle length is 2. Let a = E. Then we have E, A, E, A,. ... Cycle length is 2. Let a = F. Then we have F, A, F, A,. ... Cycle length is 2. 2. Consider the group D 3 . Note : Recall from class we labeled the three reflections F T , F R and F L based on flipping over a line through the Top vertex, Right vertex or Left vertex of the triangle. (a) Which pairs of elements of D 3 satisfy a · b = b · a ? Every element commutes with itself and with the identity, e . The only other pair that commutes is ( R 120 ,R 240 ), the 120- and 240-degree rotations.
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HW2_Solutions - Math 541 Solutions to HW#2 1 Let GL2(Z2...

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