The preceding lessons prove that p-subgroups exist, but students must also learn how to find them. The focus keyword is computing p-subgroups, because computations make Cauchy's theorem and p-group structure usable in concrete groups. In cyclic groups, the calculation is controlled by greatest common divisors. In permutation groups, the calculation is controlled by cycle structure and closure. This lesson gives a practical method before the later Sylow theorems expand the same idea to maximal p-subgroups.

Let $n$ be a positive integer and let $p$ be a prime number. The \textbf{largest p-power divisor} of $n$ is the number $p^a$ such that $p^a$ divides $n$ and $p^{a+1}$ does not divide $n$.