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Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
ISBN: 9781285463230
Textbook Problem
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A permutation matrix is a matrix that can be obtained from an identity matrix I n by interchanging the rows one or more times (that is, by permuting the rows). For n = 3 the permutation matrices are I 3 and the five matrices.

(Sec. 3.3 , # 22 c , 32 c , Sec. 3.4 , # 5 , Sec. 4.2 , # 6 )

P 1 = [ 1 0 0 0 0 1 0 1 0 ]

P 2 = [ 0 1 0 1 0 0 0 0 1 ]

P 3 = [ 0 1 0 0 0 1 1 0 0 ]

P 4 = [ 0 0 1 0 1 0 1 0 0 ]

P 5 = [ 0 0 1 1 0 0 0 1 0 ]

Given that G = { I 3 , P 1 , P 2 , P 3 , P 4 , P 5 } is a group of order 6 with respect to matrix multiplication, write out a multiplication table for G .

Sec. 3.3 , # 22 c , 32 c

22. Find the center Z ( G ) for each of the following groups G .

c. G = { I 3 , P 1 , P 2 , P 3 , P 4 , P 5 } in Exercise 35 of section 3.1.

32. Find the centralizer for each element a in each of the following groups.

c. G = { I 3 , P 1 , P 2 , P 3 , P 4 , P 5 } in Exercise 35 of section 3.1

Sec. 3.4 , # 5

5. The elements of the multiplicative group G of 3 × 3 permutation matrices are given in Exercise 35 of section 3.1 . Find the order of each element of the group.

Sec. 4.2 , # 6

6. Let G be the group of permutations matrices { I 3 , P 1 , P 2 , P 3 , P 4 , P 5 } as given in Exercise 35 of Section 3.1 .

To determine

To calculate: The multiplication table of a group G={I3,P1,P2,P3,P4,P5} of order 6.

Explanation

Given information:

A permutation matrix is a matrix that can be obtained from an identity matrix In by interchanging the rows one or more times (that is, by permuting the rows). For n = 3, the permutation matrices are I3 and the five matrices.

P1=(100001010) P2=(010100001) P3=(010001100) P4=(001010100) P5=(001100010)

Given that G={I3,P1,P2,P3,P4,P5} is a group of order 6 with respect to matrix multiplication.

Calculation:

Let G={I3,P1,P2,P3,P4,P5}.

For a multiplication table of group G, we need to calculate product of some matrices in G.

P1P1=(100001010)(100001010)=(100010001)=I3P2P3=(010100001)(010001100)=(001010100)=P4P4P5=(0<

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