The Suzuki groups are perfect #
For n ≠ 0 the generated Suzuki group TauCeti.suzukiGroup n, the subgroup of
GL₄(𝔽_(2^(2n+1))) generated by the unipotent matrices u(a, b) and the Weyl element w, is
its own derived subgroup. The restriction is necessary: TauCeti.suzukiGroup 0 is the solvable
group Sz(2).
The derived subgroup D is shown to contain every generator. The torus h(κ) of
TauCeti.GroupTheory.SpecificGroups.CFSG.Suzuki.Torus lies in the group and conjugates u(a, b)
to u(κ^θ κ⁻² a, κ^(-θ) b), θ = 2^(n+1), so its commutators with the unipotent generators are
[h(κ), u(0, b)] = u(0, (κ^(-θ) + 1) b),
[h(κ), u(a, 0)] = u((λ + 1) a, (λ + 1) a^(1+θ)), λ = κ^θ κ⁻².
For n ≠ 0 the generator field has a unit with κ^θ ≠ κ^2, so λ ≠ 1 and κ^θ ≠ 1, and the
two commutator families together with the multiplication rule u(a, b') u(0, b + b') = u(a, b)
put every u(a, b) in D. The Weyl element then follows from the rank-one Bruhat relation at
b = 1, w u(0, 1) w = u(1, 1) w u(1, 0), since w u(0, 1) w = [w, u(0, 1)] u(0, 1) is in
D once u(0, 1) is.
Nothing here concerns the centre of the group, its order, or its simplicity.
Main results #
TauCeti.Suzuki.commutator_suzukiGroup_eq_self: forn ≠ 0the Suzuki group is its own derived subgroup.TauCeti.Suzuki.isPerfect_suzukiGroup: forn ≠ 0the Suzuki group is perfect.
References #
- M. Suzuki, On a class of doubly transitive groups, Annals of Mathematics 75 (1962), 105--145.
- R. A. Wilson, The Finite Simple Groups, Springer GTM 251 (2009), §4.2.
The unipotent generators lie in the derived subgroup #
The Weyl generator lies in the derived subgroup #
Perfectness #
A Suzuki group is its own derived subgroup for n ≠ 0.
The Suzuki groups are perfect: for n ≠ 0, TauCeti.suzukiGroup n equals its
commutator subgroup. At n = 0 the group is the solvable Sz(2), and the statement fails.