# Order of Group Elements

• Oct 30th 2007, 07:42 PM
steph615
Order of Group Elements
Hi, I need help solving these problems:

Let a, b, and c be elements of a group G. Prove the following:

1. If ak =e where k is odd, then the order of a is odd.

3.The order of ab is the same as the order of ba. (Hint: if (ba)n = (baba…..b)a = e. Let (baba….b) =x then a is the inverse of x. Thus, ax =e.)

2. Let x =a1a2… an, and let y be a product of the same factors, permuted cyclically. (That is, y = akak+1…ana1…ak-1.) Then ord(x) = ord(y).

Thanks!
• Oct 30th 2007, 07:56 PM
ThePerfectHacker
Quote:

Originally Posted by steph615
1. If ak =e where k is odd, then the order of a is odd.

You probably mean \$\displaystyle a^k = e\$. If \$\displaystyle d\$ is the order of \$\displaystyle a\$ then \$\displaystyle d|k\$. So \$\displaystyle d\$ cannot be even, i.e. it must be odd.

Quote:

3.The order of ab is the same as the order of ba. (Hint: if (ba)n = (baba…..b)a = e. Let (baba….b) =x then a is the inverse of x. Thus, ax =e.)
\$\displaystyle (ab)^n = 1 \implies (ab)...(ab) = 1 \implies a (ba)^{n-1} b = 1 \implies (ba)^{n-1} = a^{-1}b^{-1}\$
But \$\displaystyle a^{-1}b^{-1} = (ba)^{-1}\$ thus \$\displaystyle (ba)^{n-1} \implies (ab)^{-1} \implies (ba)^n = 1\$.
(Why is this not a complete proof?)

Quote:

2. Let x =a1a2… an, and let y be a product of the same factors, permuted cyclically. (That is, y = akak+1…ana1…ak-1.) Then ord(x) = ord(y).
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Use induction, with the basic case covered in #2.
• Oct 30th 2007, 08:11 PM
steph615
Re
Ok, the proof for ord(ab)=ord(ba) is not complete because don't you have to show that n is the smallest positive integer? How then would I show that?