Abstract Algebra Answers

b) Let 















σ =

3 4 5 2 1

1 2 3 4 5

and 















τ =

5 3 2 4 1

1 2 3 4 5

in . S7

Write στ as a product

of disjoint permutations.

Further, is στ even? Why, or why not?

Which of the following statements are true? Give reasons for your answers, in the form

of a short proof or a counterexample.

i) ( ) M3 Z has no nilpotent elements.

Show that z(sqrt-2) is non ufd

Let a = (1 2 3 4 5

3 4 5 2 1) and b =(1 2 3 4 5

5 3 2 4 1) in S7. Write ab as a product of disjoint permutations. Further, is ab even? Why, or why not?

Is R = {[a a+b

a+b b ] |a,b belongs to Z} a subring of M2(Z)? Why or why not?

Let a = (3 4 5 2 1) and b = (5 3 2 4 1) in S7. Write ab as a product of disjoint permutations. Further, is ab even? Why or why not?

Find the permutation in S_4​such that 1→3, 2 2→1, 3→2.

There are the permutations σ=(123), τ=(12). Calculate σ(τ)=τσ (applying \tauτ first and then applying \sigmaσ)

Find the order of the element 2in the group (Z_6,+)

Find the element which belongs to the ring Z_2[x] /x^2+x+1

Let the order of element aa in the finite group be 20. Find the order of element a^6


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Find the permutation in S_4

​

such that 1→3, 2 12→1, 23→2.

Calculate 2+2 in Z_3


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