Inner and Outer Automorphism Groups
The composition of two inner automorphisms is again an inner automorphism (as mentioned above: (xa)b=xab, and with this operation, the collection of all inner automorphisms of G is itself a group, the inner automorphism group of G denoted Inn(G).
Inn(G) is a normal subgroup of the full automorphism group Aut(G) of G. The quotient group
- Aut(G)/Inn(G)
is known as the outer automorphism group Out(G). The outer automorphism group measures, in a sense, how many automorphisms of G are not inner. Every non-inner automorphism yields a non-trivial element of Out(G), but different non-inner automorphisms may yield the same element of Out(G).
By associating the element a in G with the inner automorphism ƒ(x) = xa in Inn(G) as above, one obtains an isomorphism between the quotient group G/Z(G) (where Z(G) is the center of G) and the inner automorphism group:
- G/Z(G) = Inn(G).
This is a consequence of the first isomorphism theorem, because Z(G) is precisely the set of those elements of G that give the identity mapping as corresponding inner automorphism (conjugation changes nothing).
Read more about this topic: Inner Automorphism
Famous quotes containing the words outer and/or groups:
“Take away an accident of pigmentation of a thin layer of our outer skin and there is no difference between me and anyone else. All we want is for that trivial difference to make no difference.”
—Shirley Chisholm (b. 1924)
“In America every woman has her set of girl-friends; some are cousins, the rest are gained at school. These form a permanent committee who sit on each other’s affairs, who “come out” together, marry and divorce together, and who end as those groups of bustling, heartless well-informed club-women who govern society. Against them the Couple of Ehepaar is helpless and Man in their eyes but a biological interlude.”
—Cyril Connolly (1903–1974)