Outer Automorphism Group - Out(G) For Some Finite Groups

Out(G) For Some Finite Groups

For the outer automorphism groups of all finite simple groups see the list of finite simple groups. Sporadic simple groups and alternating groups (other than the alternating group A6; see below) all have outer automorphism groups of order 1 or 2. The outer automorphism group of a finite simple group of Lie type is an extension of a group of "diagonal automorphisms" (cyclic except for Dn(q) when it has order 4), a group of "field automorphisms" (always cyclic), and a group of "graph automorphisms" (of order 1 or 2 except for D4(q) when it is the symmetric group on 3 points). These extensions are semidirect products except that for the Suzuki-Ree groups the graph automorphism squares to a generator of the field automorphisms.

Group Parameter Out(G)
Z infinite cyclic Z2 2; the identity and the map f(x) = -x
Zn n > 2 Zn× φ(n) = elements; one corresponding to multiplication by an invertible element in Zn viewed as a ring.
Zpn p prime, n > 1 GLn(p) (pn − 1)(pn − p )(pn − p2) ... (pn − pn−1)

elements
Sn n ≠ 6 trivial 1
S6 Z2 (see below) 2
An n ≠ 6 Z2 2
A6 Z2 × Z2(see below) 4
PSL2(p) p > 3 prime Z2 2
PSL2(2n) n > 1 Zn n
PSL3(4) = M21 Dih6 12
Mn n = 11, 23, 24 trivial 1
Mn n = 12, 22 Z2 2
Con n = 1, 2, 3 trivial 1

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