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 , underlying prime characteristic of residue field , and length . The size of the whole ring is , and each successive power of the unique maximal ideal has index in its predecessor.
Examples are (here ), , and Galois rings.
The Galois ring with residue field of size and length is the unique (up to isomorphism) discrete valuation ring obtained as a degree extension of , and hence generalizes both and . The Galois ring has characteristic .
Particular cases
The size of the group is and the number of conjugacy classes is . Some particular cases for small values of are given below.
Order of the group | Number of conjugacy classes | List of all possible rings | List of corresponding groups | ||
---|---|---|---|---|---|
any | 1 | the field | the general linear group , denoted . | ||
2 | 1 | 6 | 3 | field:F2 | symmetric group:S3 |
2 | 2 | 96 | 14 | ring of integers modulo 4 denoted , also | 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 , also | 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 , also | 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 , also | general linear group:GL(2,Z27) and ? |