# Proves in Group Theory

• Apr 2nd 2010, 09:55 PM
rainyice
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.
• Apr 3rd 2010, 12:15 AM
Swlabr
Quote:

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.
• Apr 3rd 2010, 01:39 PM
rainyice
Quote:

Originally Posted by Swlabr
One word hind: Induction.

Do you mean do each part of the question by induction?
• Apr 3rd 2010, 02:32 PM
Chris L T521
Quote:

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)
• Apr 4th 2010, 08:23 AM
Swlabr
Quote:

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.
• Apr 4th 2010, 09:34 AM
FancyMouse
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)