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Nontrivial Schur multipliers for SL2 over Galois rings

Ascertain whether for the Galois ring (p^l, m) the Schur multiplier H_2(SL_2((p^l, m)), Z) is isomorphic to Z/2 when p=2, m=1 and l≥2; Z/2 when p=2, m=2 and l≥1; Z/3 when p=3, m=2 and l≥1; and 0 otherwise, thereby determining precisely which Galois rings yield nontrivial H_2(SL_2((p^l, m)), Z).

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Background

After establishing vanishing results for the Schur multiplier of SL_2 over Galois rings under broad conditions (odd characteristic residue field and excluding small sizes), the authors pose a sharper classification question aiming to isolate exactly the parameter regimes (p, l, m) that produce nontrivial multipliers.

This question seeks to determine whether the nontrivial cases coincide with a small set of classical exceptions suggested by their computations and known results, providing a complete characterization within the class of Galois rings.

References

We ask whether, for a Galois ring A, the only A with non-trivial Schur multiplier of $_2(A)$ are precisely those related to the classical cases discussed in the introduction. More precisely, we ask whether for the Galois ring A=(pl, m), H_2(_2(A),Z)\simeq \begin{cases} Z/2 & \text{if $p=2$, $m=1$ and $l\geq 2$}\ Z/2 & \text{if $p=2$, $m=2$ and $l\geq 1$}\ Z/3 & \text{if $p=3$, $m=2$ and $l\geq 1$}\ 0 & \text{otherwise}\ \end{cases}? ]

— Schur multiplier of $\mathrm{SL}_2$ over finite commutative rings (2510.03946 - Mirzaii et al., 4 Oct 2025) in Remark following Corollary galois (Section 5)