Suppose $x\in G$ and $|x|=n<\infty.$ If $n=st$ for some positive integers $s$ and $t$, prove that $|x^s|=t.$
Observe that if we prove that $(x^s)^t=x^{st}$ then $1=x^n=x^{st}=(x^s)^t$ and we are done. We did not prove that $(x^s)^t=x^{st}$ in class and it is another exercise in the book. Is mathematical induction the only way to prove it?
My proof:
Statement: $(x^s)^t=x^{st}$ for all $1\leq s,t\leq n.$
Base Case: Trivially true.
Inductive Hypothesis: Assume that $(x^s)^t=x^{st}$ for all $1\leq s,t\leq n.$
Now consider (also assuming $x^{s+t}=x^sx^t$):
\begin{align} (x^{s+1})^{t+1}\\ &=(x^{s+1})^t x^{s+1}\\ &=(x^sx)^tx^sx\\ &=(x^sxx^sx\cdots x^sx)x^sx\\ &=(x^s)^tx^tx^sx\\ &=x^{st}x^{s+t+1}\\ &=x^{st+s+t+1}\\ &=x^{(s+1)(t+1)} \end{align} So by the principle of mathematical induction the statement is true.
If there are any mistakes please point them out.