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

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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 ?