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Let x belong to a group and |x| = 6. Find |x^2|, |x^3|, |x^4|, and |x^5|. Let y belong to a group and |y|=9. Find |y^i| for i=2, 3, ...,8. Do these examples suggest any relationship between the order of the power of an element and the order of the element?

Question

Let x belong to a group and |x| = 6. Find |x^2|, |x^3|, |x^4|, and |x^5|. Let y belong to a group and |y|=9. Find |y^i| for i=2, 3, ...,8. Do these examples suggest any relationship between the order of the power of an element and the order of the element? 

check_circleAnswer
Step 1

Let G be the group, e be the identity element of G and x∈G such that |x|=6 or x6=e which means the order of x is 6.

First, we need to find the order of the elements x2, x3, x4 and x5.

Step 2

Now,

Notice that
х6
-(1)
=> (x2)3
which implies that the order of x2 is 3
Similarly,
= e
=> (x3)2
which implies that the order of x3 is 2.
Now, x4, (x4) = x4x4 = x8
= x2x6
= e
= x2e (by using (1))
-(2)
(x4)2x4x2x4 (by using (2))
=> (x4)2 x2
Again, (x4)3
= x6
=> (x4)3
which impliess that the order of x4 is 3
(by using (1))
= e
help_outline

Image Transcriptionclose

Notice that х6 -(1) => (x2)3 which implies that the order of x2 is 3 Similarly, = e => (x3)2 which implies that the order of x3 is 2. Now, x4, (x4) = x4x4 = x8 = x2x6 = e = x2e (by using (1)) -(2) (x4)2x4x2x4 (by using (2)) => (x4)2 x2 Again, (x4)3 = x6 => (x4)3 which impliess that the order of x4 is 3 (by using (1)) = e

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Step 3

Again,

...
x5, (x5)2x5x5 = x10
= x4x6
= x e (by using (1))
x4 -
=>(x5)2 =
Again, (x53
-(3)
(x5)2x5 x4x5 (by using (3))
= x9
= x3x6
= x3e (by using (1))
=> (x5)3 x3
=> ((x5)3)2 (x3)2 (squaring both sides)
=> (x5)6
which implies that the order of x5 is 6.
= X
(by using (1))
help_outline

Image Transcriptionclose

x5, (x5)2x5x5 = x10 = x4x6 = x e (by using (1)) x4 - =>(x5)2 = Again, (x53 -(3) (x5)2x5 x4x5 (by using (3)) = x9 = x3x6 = x3e (by using (1)) => (x5)3 x3 => ((x5)3)2 (x3)2 (squaring both sides) => (x5)6 which implies that the order of x5 is 6. = X (by using (1))

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