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Suppose that a is a group element and a^6 = e. What are the possibilities for |a|? Provide reasons for your answer.

Question

Suppose that is a group element and a^6 = e. What are the possibilities for |a|? Provide reasons for your answer. 

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Step 1

It is known that if G be a group and a be any element of group G, then n  is said to be order of a if n is the least positive integer such that an = e, the identity element of G.

From the definition it can be concluded that if m is any positive integer such that am = e, then |a| ≤ m.

 

It is given that a is the element in the group and G be the group and a =e
Then according to the definition of the order of the element |a| must be at most 6 or a|6
Now if a1, then a' =e
a e
= e
Thus, a1 is possible case
If a 2, then a2 =e
a-(a)
(e)
3
= e
Thus, a2 is possible
case
=
help_outline

Image Transcriptionclose

It is given that a is the element in the group and G be the group and a =e Then according to the definition of the order of the element |a| must be at most 6 or a|6 Now if a1, then a' =e a e = e Thus, a1 is possible case If a 2, then a2 =e a-(a) (e) 3 = e Thus, a2 is possible case =

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Step 2
If a3, then a = e
-(e)
= e
Thus, a3 is possible case.
Now if la 4, then a* = e
a(a')(a)
- ea?
e
Thus, a4 is not possible.
help_outline

Image Transcriptionclose

If a3, then a = e -(e) = e Thus, a3 is possible case. Now if la 4, then a* = e a(a')(a) - ea? e Thus, a4 is not possible.

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