Which finite simple groups have trivial Schur multiplier? In Lie theory there are only 3 compact simple groups that have both trivial center and trivial fundamental group: $ G_2,F_4,E_8 $.
I was expecting that similarly there would be only a few finite groups which are both simple and superperfect (have trivial Schur multiplier). However it seems that $ PSL(2,2^k) $ is simple and Schur trivial for all $ k \geq 3 $. (weirdly $ PSL(2,2^2) $ is simple but has Schur multiplier $ 2 $ since it is isomorphic to $ PSL(2,5) \cong A_5 $).
This leads me to wonder, are there lots more out there...? So I'm curious
Which finite simple groups have trivial Schur multiplier?
For the same reason that the complex points of the linear algebraic groups $ G_2,F_4,E_8 $ (and thus also the compact real form) are simple centerless and have trivial fundamental group we expect the finite field points of $ G_2,F_4,E_8 $ to be simple and have trivial Schur multiplier. They are indeed all Schur trivial (with a few exceptions):

*

*$ E_8(q) $


*$ F_4(q)$ for $ q \geq 3 $


*$ G_2(q)$ for $ q \geq 5 $
(here $ q $ is any prime power)
 A: Here is the complete list of Schur trivial finite simple groups :
Nearly all the finite simple groups of Lie type for characteristic 2 are Schur trivial

*

*$ A_n(2^m) $ i.e. $ PSL(n+1,2^m) $ Schur trivial exactly when $ 2^m-1 $ is coprime to $ n+1 $, except cases $ (n=1,m=1),(n=1,m=2),(n=2,m=1),(n=2,m=2),(n=3,m=1) $ which are not Schur trivial


*$ B_n(2^m) \cong C_n(2^m) $ for $ n \geq 2 $ and all $ m $, except the cases $ n=2,m=1 $ and $ n=3, m=1 $.


*$ D_n(2^m) $ for $ n \geq 4 $ and all $ m $, except the case $ (n=4,k=1) $


*$ G_2(2^m) $ for $ m \geq 3 $


*$ F_4(2^m) $ for $ m \geq 2 $


*$ E_6(2^m) $ Schur trivial exactly when $ 2^m-1 $ is coprime to $ 3 $


*$ E_7(2^m) $ all $ m $


*$ E_8(2^m) $ all $ m $


*$ ^2A_n(2^m) $ i.e. $ PSU(n+1,2^m) $ for $ n \geq 2 $, Schur trivial exactly when $ 2^m+1 $ is coprime to $ n+1 $ except cases $ (n=3,m=1),(n=5,m=1) $


*$ ^2D_n(2^m) $ for $ n \geq 4 $ and all $ m $


*$ ^2E_6(2^m) $ , Schur trivial exactly when $ 2^m+1 $ is coprime to $ 3 $


*$ ^3D_4(2^m) $ for all $ m $


*$ ^2B_2(2^{2n+1})$ for $ n \geq 2 $


*$ ^2F_4(2^{2n+1})$ for $ n \geq 1 $, also Tits group $ ^2F_4(2)' $
For odd characteristic, some finite simple groups of Lie type are still Schur trivial, but it's a lot less. They are: (here $ q $ is an odd prime power)

*

*$ A_n(q) $ i.e. $ PSL(n+1,q) $ Schur trivial exactly when $ q-1 $ is coprime to $ n+1 $, except case (n=1,q=9) which is not Schur trivial (indeed $ PSL(2,9) \cong A_6$ )


*$ G_2(q) $ for $ q \neq 3 $


*$ F_4(q) $


*$ E_6(q) $


*$ E_8(q) $


*$ ^2A_n(q) $ i.e. $ PSU(n+1,q) $, for $ n \geq 2 $, Schur trivial exactly when $ q+1 $ is coprime to $ n+1 $, except case $ (n=3,q=3) $


*$ ^2E_6(q) $ , Schur trivial exactly when $ q+1 $ is coprime to $ 3 $


*$ ^3D_4(q) $


*Ree groups $ ^2G_2(3^{2n+1})$ for $ n \geq 1 $, also  $ ^2G_2(3)'\cong PSL(2,8) $
That is all the Schur trivial finite simple groups of Lie type.
Beyond that, of the 26 exceptional groups exactly these 13 are Schur trivial

*

*$ M_{11}, M_{23}, M_{24} $


*$ J_1,J_4 $


*$ Co_2,Co_3 $


*$ Fi_{23} $


*$ He $


*$ HN $


*$ Ly $


*$ Th $


*$ M $
Finally no simple alternating group is Schur trivial. But all the cyclic simple groups are Schur trivial

*

*all cyclic groups of prime order

A: With a few small exceptions, the Schur Multiplier of ${\rm PSL}(n,q) = A_{n-1}(q)$ is cyclic of order $\gcd(n,q-1)$. So it has trivial Schur Multiplier whenever $n$ and $q-1$ are coprime. The same applies to ${\rm PSU}(n,q)$ when $\gcd(n,q+1)=1$. There are similar but simpler conditions for the other classical groups.
The Schur Multipliers are known for all finite simple groups. They are listed, for example, in the ATLAS of finite simple groups.
