  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?

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, help_outlineImage TranscriptioncloseNotice 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 fullscreen
Step 3

Again,

... help_outlineImage Transcriptionclosex5, (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)) fullscreen

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