A formula for the number of invertible matrices over some finite ring (Q2901710)

The problem of computing the number of invertible matrices over residue rings is analyzed. Let \(M_{n}(p,k)\) and \(M^{\mathrm{inv}}_{n}(p,k)\) (\(p\) prime, \(k \in \mathbb{N}\)) be the number of all respectively all invertible \(n \times n\)-matrices over the ring \(\mathbb{Z}_{p^{k}}\). The formula NEWLINE\[NEWLINE | M^{\mathrm{inv}}_{n}(p,k) | = | M_{n}(p,k) | \prod\limits_{i=1}^{n} ( 1 - p^{-i}) NEWLINE\]NEWLINE can be established as follows. If \(k=1\) then the number of selections for \(\mathbf{a}_{i} \in \mathbb{Z}^{n}_{p}\) \((i = 1, \dots, n)\) not being any linear combination of \(\mathbf{a}_{1}, \dots, \mathbf{a}_{i-1}\) is computed. If \(k \geq 2\) then the number of sums \(B+C\) is computed, where \(B \in M^{\mathrm{inv}}_{n}(p,1)\) and \(C \in M_{n}(p,k)\). Let \(M_{n} (l)\) and \(M^{\mathrm{inv}}_{n}(l)\) \((l=p^{{\alpha}_{1}}_{1} \dots p^{{\alpha}_{m}}_{m}\) the canonical factorization) be the number of all respectively all invertible \(n \times n\)-matrices over the ring \(\mathbb{Z}_{l}\). The formula NEWLINE\[NEWLINE | M^{\mathrm{inv}}_{n}(l) | = | M_{n}(l) | \prod\limits_{j=1}^{m} \prod\limits_{i=1}^{n} ( 1 - p^{-i}_{j}) NEWLINE\]NEWLINE can be established by applying the following variant of Leng's homomorphism theorem for residue rings. For any finite non-empty set \(S\) and any pairwise prime integers \(a_{1}, \dots, a_{m} \in \mathbb{N}\) let \(F_{a_{i}}(S) = \{ f \mid f: S \rightarrow \mathbb{Z}_{a_{i}} \}\) \((i = 1, \dots, m)\), \(F(S) = \{ f \mid f: S \rightarrow \mathbb{Z}_{\prod\limits_{i=1}^{m}a_{i}} \}\), \(\widehat{F}_{a_{i}} \subseteq F_{a_{i}}(S)\) \((i = 1, \dots, m)\) and \(\widetilde{F}_{a_{i}}(S) = \{ f \in F(S) \mid f_{\bmod a_{i}} \in \widehat{F}_{a_{i}} (S) \}\) \((i = 1, \dots, m)\), where \(f_{\bmod a_{i}} (s) = f(s) (\bmod a_{i})\). Then there holds the identity NEWLINE\[NEWLINE | \bigcap\limits_{i=1}^{m}\widetilde{F}_{a_{i}}(S) | = \prod\limits_{i=1}^{m} | \widehat{F}_{a_{i}} (S) |.NEWLINE\]