If the inner automorphism group is trivial ( when a group is abelian ), the automorphism group and outer automorphism group are naturally identified; that is, the outer automorphism group does act on the group.
42.
If the inner automorphism group is trivial ( when a group is abelian ), the automorphism group and outer automorphism group are naturally identified; that is, the outer automorphism group does act on the group.
43.
More pithily : an automorphism that stabilizes transpositions is inner, and there are only two conjugacy classes of order 15 ( transpositions and triple transpositions ), hence the outer automorphism group is at most order 2.
44.
In this case the fundamental group is a free group and the outer automorphism group Out ( Fn ) is strictly larger then the image of the mapping class group via the morphism defined in the previous paragraph.
45.
In fact M 12 has two inequivalent actions on 12 points, exchanged by an outer automorphism; these are analogous to the two inequivalent actions of the symmetric group " S " 6 on 6 points.
46.
There is just one exception to this : the alternating group " A " has outer automorphism group of order 4, rather than 2 as do the other simple alternating groups ( given by conjugation by an odd permutation ).
47.
M 22 has three rank 3 permutation representations : one on the 77 hexads with point stabilizer 2 4 : A 6, and two rank 3 actions on 176 heptads that are conjugate under an outer automorphism and have point stabilizer A 7.
48.
The outer automorphism group has order 2, and the full automorphism group M 12.2 is contained in M 24 as the stabilizer of a pair of complementary dodecads of 24 points, with outer automorphisms of M 12 swapping the two dodecads.
49.
The outer automorphism group of " R " is an infinite simple group with countable many conjugacy classes, indexed by pairs consisting of a positive integer " p " and a complex " p " th root of 1.
50.
This relation can be used both ways : given an outer automorphism, one can produce new representations ( if the representation is not equal on conjugacy classes that are interchanged by the outer automorphism ), and conversely, one can restrict possible outer automorphisms based on the character table.
