𝑆𝐿⁒(2,𝔽p) has no 1 dimensional irreducible representations over 𝔽p


The group G=SL⁑(2,Fp) has no non-trivial 1 dimensional irreducible representations over Fp.


Notice that a 1 dimensional representations over 𝔽p is just a homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath:


Let ρ be as above. Then there exist an induced homomorphism for the projective special linear groupMathworldPlanetmath:


defined by ρ¯⁒(A)=ρ⁒(B)mod{±𝐼𝑑}, where B is any lift of A to SL⁑(2,𝔽p) (this is well defined because ρ⁒(-𝐼𝑑)=±𝐼𝑑). Since PSL⁑(2,𝔽p) is simple, the image of ρ¯ is trivial, and therefore, the image of ρ is contained in {±𝐼𝑑}.

However, SL⁑(2,𝔽p) does not have subgroupsMathworldPlanetmathPlanetmath of index 2 (a subgroup of index 2 is normal). For our purposes, it suffices to show that:


satisfies ρ⁒(-𝐼𝑑)=𝐼𝑑. Let S∈SL⁑(2,𝔽p) be the matrix:


Notice that S2=-𝐼𝑑 and so ρ⁒(S2)=(ρ⁒(S))2=𝐼𝑑=ρ⁒(-𝐼𝑑), as desired. ∎

Title 𝑆𝐿⁒(2,𝔽p) has no 1 dimensional irreducible representations over 𝔽p
Canonical name mathitSL2mathbbFpHasNo1DimensionalIrreducibleRepresentationsOvermathbbFp
Date of creation 2013-03-22 15:09:56
Last modified on 2013-03-22 15:09:56
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 7
Author alozano (2414)
Entry type Theorem
Classification msc 20G15