# Element structure of special linear group of degree two over a finite discrete valuation ring

## Contents

This article gives specific information, namely, element structure, about a family of groups, namely: special linear group of degree two. This article restricts attention to the case where the underlying ring is a finite discrete valuation ring.
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This article discusses the element structure of the special linear group of degree two over a finite discrete valuation ring (i.e., local principal ideal ring) with residue field having size $q$, underlying prime characteristic of residue field $p$, and length $l$. The size of the whole ring is $q^l$, and each successive power of the unique maximal ideal has index $q$ in its predecessor.

Examples are $\mathbb{Z}/(p^l\mathbb{Z})$ (here $p = q$), $\mathbb{F}_q[t]/(t^l)$, and Galois rings.

The Galois ring with residue field of size $q$ and length $l$ is the unique (up to isomorphism) discrete valuation ring obtained as a degree $\log_pq$ extension of $\mathbb{Z}/p^l\mathbb{Z}$, and hence generalizes both $\mathbb{Z}/(p^l\mathbb{Z})$ and $\mathbb{F}_q$. The Galois ring has characteristic $p^l$.

## Summary

Item Value
order of the group $q^{3l - 2}(q^2 - 1)$.
number of conjugacy classes Case $q$ odd: $q^l + 4 \frac{q^l - 1}{q - 1}$, which is $q^l + 4(q^{l-1} + q^{l-2} + \dots + 1)$
Case $q$ a power of 2: PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]