Quasisimple groups

A finite group G is quasisimple when it is perfect (G = [G, G]) and its central quotient G / Z(G) is simple. Equivalently, G is a perfect central extension — a cover — of a non-abelian simple group. These groups are the building blocks (the components) from which the structure theory of finite groups is assembled.

Definition

G is quasisimple if both hold:

Perfect: G = [G, G] (see commutator, perfect).
Simple central quotient: G / Z(G) is simple (see center).

The quotient S = G / Z(G) is automatically non-abelian simple: G perfect forces S perfect, and the only abelian simple groups — the Cₚ — are not perfect. So G sits in a central extension

1 → Z(G) → G → S → 1,     S = G/Z(G) non-abelian simple,

with G perfect. Any such G is called a cover of S.

Universal (Schur) cover

Each non-abelian finite simple group S has a universal perfect central extension Ŝ, unique up to isomorphism, whose center is the Schur multiplier

Z(Ŝ) ≅ M(S) = H₂(S, ℤ)

(see homology). Every quasisimple G with G / Z(G) ≅ S is a quotient of Ŝ:

covers ↔ subgroups N ≤ M(S), via G ≅ Ŝ / N;
the resulting center is Z(G) ≅ M(S) / N;
N = M(S) recovers S itself (trivial center); N = {1} gives the universal cover Ŝ.

Since M(S) is finite abelian, its subgroups and quotients have the same isomorphism types, so — up to isomorphism — the possible centers Z(G) range exactly over the subgroups of M(S).

Components and the layer

For an arbitrary finite group G:

a component is a quasisimple subnormal subgroup (a member of a chain H ◁ ⋯ ◁ G);
distinct components commute elementwise;
the layer E(G) is the subgroup they generate — a central product of the components, and a characteristic subgroup of G.

Together with the Fitting subgroup F(G), the layer forms the generalized Fitting subgroup F*(G) = E(G) F(G), which satisfies C_G(F*(G)) ⊆ F*(G) — it contains its own centralizer. This self-centralizing property makes F*(G) the cornerstone of the general structure theory of finite groups.

Examples

Every non-abelian finite simple group is quasisimple: Z(G) = {1}, so G / Z(G) = G is simple, and G is perfect. E.g. A₅, PSL(2,𝔽₇).
SL(2,𝔽₅) — order 120, center {±I} ≅ C₂, with SL(2,𝔽₅) / {±I} ≅ PSL(2,𝔽₅) ≅ A₅. It is the double cover 2.A₅: quasisimple but not simple. Since M(A₅) = C₂, it is the universal cover — the only covers of A₅ are A₅ and SL(2,𝔽₅).
SL(2,𝔽_q), q ≥ 4 — quasisimple, covering PSL(2,𝔽_q). For odd q the center is C₂ and SL(2,𝔽_q) is the double cover 2.PSL(2,𝔽_q); e.g. SL(2,𝔽₉) ≅ 2.A₆, order 720.
2.A₆. The multiplier M(A₆) = C₆ is exceptional (for n ≥ 5, n ≠ 6, 7, one has M(Aₙ) = C₂), so A₆ has four covers — A₆, 2.A₆, 3.A₆, 6.A₆ — one for each subgroup of C₆. Here 2.A₆ ≅ SL(2,𝔽₉) and 6.A₆ is the universal cover.

Non-examples

GL(2,𝔽₅) — not perfect. The determinant GL(2,𝔽₅) → 𝔽₅ˣ ≅ C₄ maps onto an abelian group, so [GL(2,𝔽₅), GL(2,𝔽₅)] ≠ GL(2,𝔽₅). (Its subgroup SL(2,𝔽₅) is quasisimple.)
A₅ × A₅ — perfect, but its central quotient is not simple. Both factors are centerless, so Z = {1} and the quotient is A₅ × A₅ itself, which is not simple. It has two components (the two A₅ factors), with layer E = A₅ × A₅.
SL(2,𝔽₃) — a central extension of A₄, but A₄ is not simple and SL(2,𝔽₃) is not perfect (abelianization C₃).