Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 11F (Number Theory)

The reduction map sends $H_k$ onto $H_{k-1}$ with kernel of order $p$, so $|H_k|=p^{k-1}$. More explicitly, the lifting-the-exponent calculationshows that $1+p$ has order $p^{k-1}$ modulo $p^k$. Hence $H_k=⟨1+p⟩$ is cyclic of that order.

For the matrix claim, suppose $A\ne I$ and choose $m$ maximal such that $A=I+p^{m}B$, so some entry of $B$ is nonzero modulo $p$. Replacing $A$ by a suitable power, we may assume its order is a prime $q$. If $q\ne p$, binomial expansion modulo $p^{m+1}$ givesIf $q=p$, expansion modulo $p^{m+2}$ givesbecause $p$ is odd. Both contradict $A^q=I$, so $A=I$.

For $p=2$, $r=1$ fails because $-I=I+2(-I)$ has order two. If $r=2$, the same argument works; in the order-two case, for maximal $m\ge 2$,since the parenthesis is nonzero modulo two. Thus the smallest value is $r=2$.