Finite simple groups

A finite simple group is a nontrivial finite group whose only normal subgroups are the identity subgroup and the whole group. The classification says that every such group is cyclic of prime order, alternating, of Lie type, or sporadic. This atlas provides navigation through all 18 infinite families, the 26 sporadic groups, and the Tits small-parameter exception. Different family parameters can describe isomorphic groups; the identifications below are part of the classification, not additional groups. Wilson’s introductory lecture gives the classification statement and the alternating-group examples.

This is a classification of simple groups. An arbitrary finite symmetry group may have normal subgroups, central extensions, semidirect products or direct products that are essential to its representations. Use the individual reference records when your physical symmetry is one of those groups.

Prime cyclic and alternating groups

Family	Parameter range
Prime cyclic Cₚ	p is prime; these are exactly the abelian finite simple groups.
Alternating Aₙ	n ≥ 5; even permutations of n symbols.

The trivial group is excluded. A₄ belongs in the general finite-group reference, but not in the simple alternating family. A₅ and A₆ also occur under projective linear names.

The six classical Lie-type families

Here q = pᶠ, with p prime and f a positive integer. Projective quotients and the Ω subgroup in the orthogonal cases are essential to the group identity.

Family	Range used in this atlas
Linear PSL(n,q)	n ≥ 2; exclude (n,q) = (2,2), (2,3).
Unitary PSU(n,q)	n ≥ 3; exclude (3,2). Matrices are over F(q²).
Symplectic PSp(2n,q)	n ≥ 2; exclude (2,2).
Odd orthogonal PΩ(2n+1,q)	n ≥ 3 and q odd.
Even orthogonal PΩ⁺(2n,q)	n ≥ 4; split, or plus, type.
Even orthogonal PΩ⁻(2n,q)	n ≥ 4; nonsplit, or minus, type.

These rank bounds avoid repetitions: low-dimensional orthogonal groups identify with linear, unitary or symplectic groups. In even characteristic, the odd-dimensional orthogonal groups identify with symplectic groups. The family pages explain those cases. See Wilson’s classical-group notes, §§3.2–3.5, and GAP’s classical-group conventions.

The ten exceptional Lie-type families

The superscripts on twisted groups belong to their mathematical names; they are not powers of the group. The following table states all parameter restrictions used here.

Family	Parameter range
G₂(q)	Prime power q ≥ 3.
F₄(q)	Any prime power q ≥ 2.
E₆(q)	Any prime power q ≥ 2.
E₇(q)	Any prime power q ≥ 2.
E₈(q)	Any prime power q ≥ 2.
²E₆(q)	Any prime power q ≥ 2.
³D₄(q)	Any prime power q ≥ 2.
Suzuki ²B₂(q)	q = 2^(2m+1), m ≥ 1.
Small Ree ²G₂(q)	q = 3^(2m+1), m ≥ 1.
Large Ree ²F₄(q)	q = 2^(2m+1), m ≥ 1.

Wilson’s exceptional-group lecture discusses these families and their small cases. For constructions of Suzuki and Ree groups, see his original research paper.

Sporadic groups and the Tits convention

The sporadic collection contains precisely 26 groups: five Mathieu groups; four Janko groups; three Conway groups; Fi₂₂, Fi₂₃ and Fi₂₄′; HS, McL, He, Ru, Suz, O’N, HN, Ly and Th; the Baby Monster B; and the Monster M. Individual entries give exact orders and links to the ATLAS representation records.

The Tits group ²F₄(2)′ is listed separately as a Lie-type small case. It is the simple derived subgroup of index two in the nonsimple group ²F₄(2), and is not counted as a 27th sporadic group here. Its order is 17971200. See the ATLAS Tits record.

Recognize the same group under different names

Identification	Reference or destination
PSL(2,4) ≅ PSL(2,5) ≅ A₅	A₅
PSL(2,7) ≅ PSL(3,2)	PSL(2,7)
PSL(2,9) ≅ PSp(4,2)′ ≅ A₆	A₆
PSL(4,2) ≅ A₈	Alternating family
PSU(4,2) ≅ PSp(4,3)	Unitary family
G₂(2)′ ≅ PSU(3,3)	G₂ family
²G₂(3)′ ≅ PSL(2,8)	Small Ree family

These identifications are collected in Craven’s paper, §7, the seven-item list before Proposition 24. A prime denotes a derived subgroup: dropping it in the last two rows changes the group.

Choose the representation question

A finite group of Lie type is a finite-field object. A compact Lie group bearing the same root-system label is a different object, with a different representation problem. Keep the group, coefficient field and characteristic together; the GAP classification interface makes the finite-group parameters explicit.

Classification coverage does not promise a complete local character table or an executable calculation. Reference entries with an irrep summary identify its category as ordinary complex and state whether the list is complete. Modular irreducibles, projective representations and representations of central covers require their own specifications. ATLAS links and exact-order records support further investigation; calculator availability is stated separately on each entry.