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Field:F2
From Groupprops
Contents |
Definition
This field, denoted F2 or GF(2), is the unique field (up to isomorphism) of two elements.
Facts
- For any natural number n, the projective general linear group, projective special linear group, special linear group, and general linear group are all the same.
Related groups
| Group functor | Value | GAP ID |
|---|---|---|
| additive group | cyclic group:Z2 | (2,1) |
| multiplicative group | trivial group | (1,1) |
| general affine group of degree one | cyclic group:Z2 | (2,1) |
| general linear group of degree two | symmetric group:S3 | (6,1) |
| special linear group of degree two | symmetric group:S3 | (6,1) |
| projective general linear group of degree two | symmetric group:S3 | (6,1) |
| projective special linear group of degree two | symmetric group:S3 | (6,1) |
| outer linear group of degree two | direct product of S3 and Z2 | (12,4) |
| general affine group of degree two | symmetric group:S4 | (24,12) |
| projective special linear group of degree three | projective special linear group:PSL(3,2) | (168,42) |
| projective special linear group of degree four | alternating group:A8 | The order is 20160, no GAP ID. |
GAP implementation
The field can be defined using GAP's GF function as:
GF(2)
It can also be defined as a ring using GAP's ZmodnZ function as:
ZmodnZ(2)
