Group of Lie Type - Small Groups of Lie Type

Small Groups of Lie Type

In general the finite group associated to an endomorphism of a simply connected simple algebraic group is the universal central extension of a simple group, so is perfect and has trivial Schur multiplier. However some of the smallest groups in the families above are either not perfect or have a Schur multiplier larger than "expected".

Cases where the group is not perfect include

• A₁(2) = SL(2, 2) Solvable of order 6 (the symmetric group on 3 points)
• A₁(3) = SL(2, 3) Solvable of order 24 (a double cover of the alternating group on 4 points)
• 2A₂(4) Solvable
• B₂(2) Not perfect, but is isomorphic to the symmetric group on 6 points so its derived subgroup has index 2 and is simple of order 360.
• 2B₂(2) = Suz(2) Solvable of order 20 (a Frobenius group)
• 2F₄(2) Not perfect, but the derived group has index 2 and is the simple Tits group.
• G₂(2) Not perfect, but the derived group has index 2 and is simple of order 6048.
• 2G₂(3) Not perfect, but the derived group has index 3 and is the simple group of order 504.

Some cases where the group is perfect but has a Schur multiplier that is larger than expected include:

• A₁(4) The Schur multiplier has an extra Z/2Z, so the Schur multiplier of the simple group has order 2 instead of 1.
• A₁(9) The Schur multiplier has an extra Z/3Z, so the Schur multiplier of the simple group has order 6 instead of 2.
• A₂(2) The Schur multiplier has an extra Z/2Z, so the Schur multiplier of the simple group has order 2 instead of 1.
• A₂(4) The Schur multiplier has an extra Z/4Z × Z/4Z, so the Schur multiplier of the simple group has order 48 instead of 3.
• A₃(2) The Schur multiplier has an extra Z/2Z, so the Schur multiplier of the simple group has order 2 instead of 1.
• B₃(2) = C₃(2) The Schur multiplier has an extra Z/2Z, so the Schur multiplier of the simple group has order 2 instead of 1.
• B₃(3) The Schur multiplier has an extra Z/3Z, so the Schur multiplier of the simple group has order 6 instead of 2.
• D₄(2) The Schur multiplier has an extra Z/2Z × Z/2Z, so the Schur multiplier of the simple group has order 4 instead of 1.
• F₄(2) The Schur multiplier has an extra Z/2Z, so the Schur multiplier of the simple group has order 2 instead of 1.
• G₂(3) The Schur multiplier has an extra Z/3Z, so the Schur multiplier of the simple group has order 3 instead of 1.
• G₂(4) The Schur multiplier has an extra Z/2Z, so the Schur multiplier of the simple group has order 2 instead of 1.
• 2A₃(4) The Schur multiplier has an extra Z/2Z, so the Schur multiplier of the simple group has order 2 instead of 1.
• 2A₃(9) The Schur multiplier has an extra Z/3Z × Z/3Z, so the Schur multiplier of the simple group has order 36 instead of 4.
• 2A₅(4) The Schur multiplier has an extra Z/2Z × Z/2Z, so the Schur multiplier of the simple group has order 12 instead of 3.
• 2E₆(4) The Schur multiplier has an extra Z/2Z × Z/2Z, so the Schur multiplier of the simple group has order 12 instead of 3.
• 2B₂(8) The Schur multiplier has an extra Z/2Z × Z/2Z, so the Schur multiplier of the simple group has order 4 instead of 1.

There is a bewildering number of "accidental" isomorphisms between various small groups of Lie type (and alternating groups). For example, the groups SL(2, 4), PSL(2, 5), and the alternating group on 5 points are all isomorphic.

For a complete list of these exceptions see the list of finite simple groups. Many of these special properties are related to certain sporadic simple groups.

Alternating groups sometimes behave as if they were groups of Lie type over the field with one element. Some of the small alternating groups also have exceptional properties. The alternating groups usually have an outer automorphism group of order 2, but the alternating group on 6 points has an outer automorphism group of order 4. Alternating groups usually have a Schur multiplier of order 2, but the ones on 6 or 7 points have a Schur multiplier of order 6.