# Thread: Proves in Group Theory

1. ## Proves 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.

2. Originally Posted by rainyice
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.
One word hind: Induction.

3. Originally Posted by Swlabr
One word hind: Induction.

Do you mean do each part of the question by induction?

4. Originally Posted by Swlabr
One word hind: Induction.
You don't necessarily need induction for any of these (or so I think -- I know its overkill for (a)!). For question (a), just consider the two cases where a has infinite order and where a has finite order (|a|=n)

5. Originally Posted by Chris L T521
You don't necessarily need induction for any of these (or so I think -- I know its overkill for (a)!). For question (a), just consider the two cases where a has infinite order and where a has finite order (|a|=n)
Sure, but it is the most obvious way and works for all parts of the question.

6. 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)