PGammaL(2,4) is isomorphic to S5
From Groupprops
This article gives a proof/explanation of the equivalence of multiple definitions for the term symmetric group:S4
View a complete list of pages giving proofs of equivalence of definitions
Statement
The projective semilinear group of degree two over field:F4 (the field of four elements) is isomorphic to symmetric group:S5.
In symbols:
.
Related facts
Similar facts
- PGL(2,5) is isomorphic to S5
- PSL(2,5) is isomorphic to A5
- PGL(2,4) is isomorphic to A5
- PGL(2,3) is isomorphic to S4
- PSL(2,3) is isomorphic to A4
Facts used
- Equivalence of definitions of size of projective space
- Order formulas for linear groups of degree two
Proof
Step no. | Assertion/construction | Facts used | Previous steps used | Explanation |
---|---|---|---|---|
1 | For any field ![]() ![]() ![]() ![]() ![]() ![]() |
Follows from definitions. | ||
2 | For ![]() ![]() ![]() ![]() ![]() |
Fact (1) | ||
3 | For ![]() ![]() ![]() ![]() ![]() |
Fact (2) | ||
4 | For ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Steps (2), (3) | Plug in and evaluate. | |
5 | For ![]() ![]() ![]() ![]() |
Steps (1), (4) | By Steps (1) and (4), we get an injective homomorphism from ![]() ![]() |