Prove that in any group, an element and its inverse have the same order.

Khadija Wells 2020-11-20 Answered
Prove that in any group, an element and its inverse have the same order.
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Elberte
Answered 2020-11-21 Author has 95 answers

Proof:
Let x be a element in a group and x1 be its inverse.
Assume o(x)=mando(x1)=n
It is know that if ap=e, then o(a)=p
consider, o(x)=m and simplify as shown below
xm=e
(xm)1=e1
xm=e
o(x1)m
nm (1)
Now consider o(x1)=n and simplify.
(x1)n=e
xn=e
(xn)1=e1
xn=e
o(x)n
mn (2)
From equation (1) and (2), m=n
Hence proved.

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Jeffrey Jordon
Answered 2021-10-07 Author has 2047 answers

Step 1

let G be a group and a an element of G with order n (an=e),so

(a1)n=an=(an)1=e1=e

therefore 

order(a1)order(a)

Step 2

 

by the same argument we can say:

order(a)=order((a1)1)order(a1)  we conclude that :

order(a)order(a1)

Finally we conclude that: order(a)=order(a1)

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