Answer to Question #107871 in Abstract Algebra for maria

Question

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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Answer 1

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 $a^6$ .