en.wikipedia.org/wiki/Projective_linear_group
1 correction found
The associated extensions SL(n, q) → PSL(n, q) are covering groups of the alternating groups (universal perfect central extensions) for A4, A5, by uniqueness of the universal perfect central extension;
This is wrong for A4. Universal perfect central extensions exist only for perfect groups, but A4 is not perfect, and its double cover SL(2,3) is not perfect either.
Full reasoning
As written, this sentence says the extensions corresponding to both A4 and A5 are "universal perfect central extensions." That is correct for A5 (its double cover is the binary icosahedral group (SL(2,5))), but not for A4.
Why not?
- A group admits a universal central extension only when the group is perfect.
- A4 is not perfect: its commutator subgroup is (V_4 \cong C_2^2), so its abelianization is (A_4/V_4 \cong C_3).
- The corresponding double cover SL(2,3) is also not perfect: its commutator subgroup is (Q_8), and the quotient by that commutator subgroup is (C_3).
So the extension (SL(2,3) \to A_4) is a covering group in the small-group sense, but it is not a universal perfect central extension. The universal-perfect-central-extension language applies to the A5 case, not the A4 case.
3 sources
- Existence criterion for universal central extensions - Alphabeta Math
Statement: A group G admits a universal central extension if and only if G is perfect.
- A4 - GroupNames
Commutator G' = C_2^2; G/G' = C_3 ... A4 ... also denoted L_2(3) (L=PSL).
- SL(2,3) - GroupNames
Centre Z = C_2; G/Z = A_4. Commutator G' = Q_8; G/G' = C_3 ... non-abelian, soluble.