1. ## Abstract Algebra-Groups

puzzled by this problem. Suppose that a is a group element and a^(6)=e. What are the possibities for |a|?
Intuitively I want to say that the answer is 6 because of the definition of order of an element.
However another wants to say infinite...
Suggestions?

2. Originally Posted by RoboMyster5
puzzled by this problem. Suppose that a is a group element and a^(6)=e. What are the possibities for |a|?
Intuitively I want to say that the answer is 6 because of the definition of order of an element.
However another wants to say infinite...
Suggestions?
The order of an element, |a| , is defined as the smallest possible integer n such that $a^n = e$ . Immediately you should notice that this rules out infinity as a possiblity as we're given that $a^6=e$, and clearly 6 < infinity .

Since we're given $a^6=e$ it must be that |a| divides 6 . This gives the possibilites:

1) |a| = 1
2) |a| = 2
3) |a| = 3
4) |a| = 6

You can test to see if our original equation is satisfied:

Suppose |a|= 2, then is still true that $a^6 = e$ ? Well, $a^6 = (a^2)^3$ and we know that |a| = 2 and so $a^2 = e$

therefore $a^6=e^3=e$ . So it still is true. If you're not convinced you can repeat this for the other cases.

Hope this helps.

Pomp