# Changes

# Subgroups of the form $\langle a^d \rangle$, where $d | n$. There is one such subgroup for each $d$. The total number of such subgroups is $\tau(n)$ or $\sigma_0(n)$, i.e., the [[number:divisor count function|number of positive divisors]] of $n$.# Subgroups of the form $\langle a^d, a^rx\rangle$ where $d | n$ and $0 \le r < d$. There are thus $d$ such subgroups for each such divisor $d$. The total number of such subgroups is $\sigma(n)$ or $\sigma_1(n)$, i.e., the [[number:divisor sum function|sum of positive divisors]] of $n$.