# Abstract Algebra-Groups

• Sep 29th 2009, 02:33 PM
RoboMyster5
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?
• Sep 30th 2009, 12:39 PM
pomp
Quote:

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 \$\displaystyle a^n = e\$ . Immediately you should notice that this rules out infinity as a possiblity as we're given that \$\displaystyle a^6=e\$, and clearly 6 < infinity .

Since we're given \$\displaystyle 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 \$\displaystyle a^6 = e\$ ? Well, \$\displaystyle a^6 = (a^2)^3\$ and we know that |a| = 2 and so \$\displaystyle a^2 = e\$

therefore \$\displaystyle 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