TSSFL TECHNOLOGY STACK

Posted: **Wed Jan 06, 2016 11:44 am**

We provide these examples to help you get started.

Problem 1. Let $G$ be a group, $a, \ b \in G$ such that $a^5 = e, \ b \ne e, \ b^2 = aba^{-1}$ . Find the order of $b, \ |b|$ .

Solution

From $b^2 = aba^{-1} \\ \\ b^4 = (b^2)^2 = (aba^{-1})(aba^{-1}) = ab^2a^{-1}$ .

Proceeding in this manner (You should prove this with a lemma), we find that $b^{2k} = ab^{k}a^{-1}$ , $k$ positive integer.

Thus

$b^4 = ab^2a^{-1} = a(b^2)a^{-1} = a(aba^{-1})a^{-1} = a^2ba^{-2}$

$b^8 = ab^4a^{-1} = a(b^4)a^{-1} = a(a^2ba^{-2})a^{-1} = a^3ba^{-3}$

$b^{16} = ab^8a^{-1} = a(a^3ba^{-3})a^{-1} = a^4ba^{-4}$

$b^{32} = ab^{16}a^{-1} = a(b^{16})a^{-1} = a(a^4ba^{-4})a^{-1} = a^5ba^{-5}$

But $a^5 = e = a^{-5}$

So $b^{32} = b \ \implies \ b^{31} = e$ , and therefore, $|b| = 31$ .

Problem 2. Let $G$ be a group, $a, \ b \in G$ such that $a^5 = e, \ b \ne e, \ b^2 = aba^{-1}$ . Prove that $|b| = 31$ .

Proof

We have seen that $b^{2k} = ab^{k}a^{-1}$ , for $k$ positive integer.

We are looking for the smallest integer, $k$ such that $b^k = e$

Now

$b^{32} = b^{2(16)} = ab^{16}a^{-1} = a(ab^{8}a^{-1})a^{-1} = a^2(ab^4a^{-1})a^{-2} = a^3(ab^2a^{-1})a^{-3} = a^4(aba^{-1})a^{-4} = a^5ba^{-5}$

But $a^5 = e = a^{-5}$

So $b^{32} = b \ \implies \ b^{31} = e$ , and therefore, $|b| = 31$ .

TSSFL TECHNOLOGY STACK

Order of an Element in a Group - Solved Examples

Order of an Element in a Group - Solved Examples