# Element structure of general 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: general linear group of degree two.
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This article discusses the element structure of the general 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 $p^l$.

## Particular cases

The size of the group is $q^{4l - 3}(q - 1)(q^2 - 1)$ and the number of conjugacy classes is $q^{2l} - q^{l-1}$. Some particular cases for small values of $l$ are given below.

$q$ $l$ Order of the group Number of conjugacy classes List of all possible rings List of corresponding groups
any 1 $(q^2 - 1)(q^2 - q) = q(q-1)(q^2 - 1)$ $q^2 - 1$ the field $\mathbb{F}_q$ the general linear group $GL_2(\mathbb{F}_q)$, denoted $GL(2,q)$.
2 1 6 3 field:F2 symmetric group:S3
2 2 96 14 ring of integers modulo 4 denoted $\mathbb{Z}_4$, also $\mathbb{F}_2[t]/(t^2)$ general linear group:GL(2,Z4) (for both rings, i.e., although the rings are not isomorphic, the corresponding general linear groups are isomorphic)
2 3 1536 60 ring of integers modulo 8 denoted $\mathbb{Z}_8$, also $\mathbb{F}_2[t]/(t^3)$ general linear group:GL(2,Z8) and ?
3 1 48 8 field:F3 general linear group:GL(2,3)
3 2 3888 78 ring of integers modulo 9 denoted $\mathbb{Z}_9$, also $\mathbb{F}_3[t]/(t^2)$ general linear group:GL(2,Z9) (for both rings, i.e., although the rings are not isomorphic, the corresponding general linear groups are isomorphic)
3 3 314928 720 ring of integers modulo 27, denoted $\mathbb{Z}_{27}$, also $\mathbb{F}_3[t]/(t^3)$ general linear group:GL(2,Z27) and ?