Answer to Question #107871 in Abstract Algebra for maria

Question #107871
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

1
Expert's answer
2020-04-06T17:49:41-0400

The given permutation are "\\sigma=(123) \\ and \\ \\tau=(12)" .

"\\therefore \\sigma(\\tau)=(123)(12)=(13)" .

1. The given group is "\\Z_6" under addition madulo 6.

We known that "\\Z_6" is a cyclic group of order 6 generated by 1 .

i,e,"\\Z_6=<1>."

"So, \\ O(2)=" order of 2 "=1^2=(1+1)=\\frac{O(1)}{gcd(2,O(1)}=\\frac{6}{gcd(2,6)}" .

"\\therefore O(2)=3" .

2. Let the given ring is

"R=\\frac{\\Z_2[x]}{<x^2+x+1>}"

"\\therefore \\ R=\\{f(x)+<x^2+x+1>:f(x)\\in\\Z_2[x]\\}"

"=\\{ax+b+<x^2+x+1>:ax+b\\in\\Z_2[x]\\}"

"=\\{ 0+<x^2+x+1>,x+<x^2+x+1>,x+1+<x^2+x+1>,1+<x^2+x+1>\\}"


"3. \\" "Let \\ G" be a finite group and "a\\in G" such that "O(a)=20=" order of "a" .

"\\therefore \\ <a>" is a cyclic subgroup of "G" ,whose order is 20 and "a^6\\in<a>."

Hence ,"O(a^6)=\\frac{O(a)}{gcd(6,O(a)}=\\frac{20}{gcd(6,20)}=\\frac{20}{2}=10."

Where "O(a^6)" represent oder of "a^6" .


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