Criterion for $𝑎 𝑖 = 𝑎 𝑗$ in a group

For a group element $𝑎 \in 𝐺$ , $𝑎 𝑖 = 𝑎 𝑗$ iff either: #m/thm/group

$| 𝑎 | = \infty$ and $𝑖 = 𝑗$
$| 𝑎 | \in ℕ$ and $𝑛 ∣ (𝑖 - 𝑗)$

Proof

If $𝑎$ has infinite order there exists no nonzero $𝑛$ such that $𝑎 𝑛 = 𝑒$ , and since $𝑎 𝑖 = 𝑎 𝑗$ implies $𝑎 𝑖 - 𝑗 = 𝑒$ , it follows $𝑖 = 𝑗$ .

If $| 𝑎 | = 𝑛$ then we again have the implication $𝑎 𝑖 - 𝑗 = 𝑒$ . By the division algorithm $𝑖 - 𝑗 = 𝑝 𝑞 + 𝑟$ , with $0 \leq 𝑟 < 𝑛$ . Then $𝑒 = 𝑎 𝑖 - 𝑗 = 𝑎 𝑝 𝑛 + 𝑟 = 𝑎 𝑟 = 𝑒$ , and since $𝑛$ is the lowest positive integer such that $𝑎 𝑛 = 𝑒$ , it follows that $𝑟 = 0$ . Hence $𝑛 ∣ (𝑖 - 𝑗)$ .

Corollary

It immediately follows that $𝑎 𝑘 = 𝑎$ implies $| 𝑎 |$ divides $𝑘$ .

#state/tidy | #lang/en | #SemBr

Criterion for 𝑎𝑖 =𝑎𝑗 in a group

Corollary

Criterion for $𝑎 𝑖 = 𝑎 𝑗$ in a group