Relationship between $| 𝑎 𝑏 |$ and $| 𝑎 | | 𝑏 |$

Given a finite group $𝐺$ , and elements $𝑎, 𝑏 \in 𝐺$ such that $𝑎 𝑏 = 𝑏 𝑎$ , then $| 𝑎 𝑏 |$ divides $| 𝑎 | | 𝑏 |$ . #m/thm/group

Proof

Let $| 𝑎 | = 𝑚$ and $| 𝑏 | = 𝑛$ . Then $(𝑎 𝑏) 𝑚 𝑛 = (𝑎 𝑚) 𝑛 (𝑏 𝑛) 𝑚 = 𝑒$ which implies $| 𝑎 𝑏 |$ divides $𝑚 𝑛$ by Corollary.

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

Relationship between |𝑎𝑏| and |𝑎||𝑏|