Let a and b be elements of a finite group G.

a. Prove that a and a^(-1) have the same order.

b. Prove that a and b*a*b^(-1) have the same order.

c. Prove that a*b and b*a have the same order.

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- Apr 2nd 2010, 09:55 PMrainyiceProves in Group Theory
Let a and b be elements of a finite group G.

a. Prove that a and a^(-1) have the same order.

b. Prove that a and b*a*b^(-1) have the same order.

c. Prove that a*b and b*a have the same order. - Apr 3rd 2010, 12:15 AMSwlabr
- Apr 3rd 2010, 01:39 PMrainyice
- Apr 3rd 2010, 02:32 PMChris L T521
- Apr 4th 2010, 08:23 AMSwlabr
- Apr 4th 2010, 09:34 AMFancyMouse
Actually induction is not necessary. To show that "x" has the same order of "y", prove o(y)<=o(x) first, and then next o(x)<=o(y). Fortunately each one of the three cases is symmetric, i.e. you don't need to do anything to prove o(x)<=o(y) since you already got o(y)<=o(x)