# examples of outer automorphism group

It is easy to understand that ${\rm Out}\mathbb{Z}={\rm Aut}\mathbb{Z}=\mathbb{Z}/2\mathbb{Z}$, since $\mathbb{Z}$ is abelian and there are no inner-automorphisms, save the trivial one.

Also, it is known that ${\rm Out}SL(2,\mathbb{Z})=\mathbb{Z}/2\mathbb{Z}$

Another example is that, at least for orientable surfaces, the extended mapping class group (or the zeroth homeotopy group) of a surface $F$ is related to its fundamental group via ${\cal{M}^{*}}(F)={\rm Out}(\pi_{1}(F))$.

1. L. K. Hua, I Reiner , Trans Amer. Math. Soc. 71 (1951), 331-348.

2. H. Zieschang, E. Vogt, H. D. Coldewey, Surfaces and planar discontinuous groups, L.N.M. 875 (1981) Springer-Verlag.

Title examples of outer automorphism group ExamplesOfOuterAutomorphismGroup 2013-03-22 16:30:25 2013-03-22 16:30:25 juanman (12619) juanman (12619) 8 juanman (12619) Example msc 20F28