I have two problems I would like to discuss.(adsbygoogle = window.adsbygoogle || []).push({});

1.For any group G prove that the set of inner automorphisms J(G) is a normal subgroup of the set of automorphisms A(G).

Let A be an automorphism of G. Let [tex]T_{g}[/tex] be an inner automorphism, i.e.

[tex]xT_{g}=g^{-1}xg[/tex]

The problem can be reduced to the question whether the following equality is true:

[tex]xAT_{g}=xT_{g}A[/tex]

Then expanding using [tex]xT_{g}=g^{-1}xg[/tex] we have:

[tex]gxAg^{-1}=gxg^{-1}A[/tex]

However I am having trouble proving this equality. Attempting to use the definition of normal more directly also did not work, i.e. showing that

[tex]AT_{g}A^{-1}[/tex] is in J(G).

2.Let G = {e, a, b, ab} where [tex]a^2=b^2=e[/tex] and ab = ba. Determine the set of automorphisms A(G).

This problem could be handled easily using brute force but I would like some way to narrow down the possible automorphisms. My attempts to solve the problem so far have resulted in way too many calculations.

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# Homework Help: Abstract Algebra - automorphism

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