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Induced Automorphisms in extension theory??

Hi,

I'm new to TSR so if I've put this in the wrong place or it's already been discussed sorry.

I'm reading up on extension theory for an upcoming project and came across this...

The transformation of A by $g_{\alpha}$ induces an automorphism in A. We denote the image of an element $a$ of A under the isomorphism by $a^{\alpha}$;

${(g_{\alpha})^{-1}}*a*g_{\alpha} = a^{\alpha}$

Can someone explain why this is the case please?

A is a group and $g_{\alpha}$ are elements of the extension, G, of A by B such that $g_{\alpha}*A\rightarrow \alpha$ is the isomorphism between G/A and B

sorry if this doesn't make much sense. Any help would be much appreciated.
Thanks.
Original post by jalw500
Hi,

I'm new to TSR so if I've put this in the wrong place or it's already been discussed sorry.

I'm reading up on extension theory for an upcoming project and came across this...

The transformation of A by $g_{\alpha}$ induces an automorphism in A. We denote the image of an element $a$ of A under the isomorphism by $a^{\alpha}$;

${(g_{\alpha})^{-1}}*a*g_{\alpha} = a^{\alpha}$

Can someone explain why this is the case please?

A is a group and $g_{\alpha}$ are elements of the extension, G, of A by B such that $g_{\alpha}*A\rightarrow \alpha$ is the isomorphism between G/A and B

sorry if this doesn't make much sense. Any help would be much appreciated.
Thanks.


You will get more of a response in the maths forum. @Zacken any chance of moving it for the OP?

BTW, studentroom uses "" tags rather than $ for latex.
Reply 2
Avatar for Zacken
Zacken
22
Original post by Gregorius
You will get more of a response in the maths forum. @Zacken any chance of moving it for the OP?

BTW, studentroom uses "" tags rather than $ for latex.


Yep, thanks for the tag. Will be moved within an hour, I can't do it myself sincr it's not in study help but I've notified the relevant people. :-)
Original post by jalw500

The transformation of A by gαg_{\alpha} induces an automorphism in A. We denote the image of an element aa of A under the isomorphism by aαa^{\alpha};

(gα)−1∗a∗gα=aα \displaystyle{(g_{\alpha})^{-1}}*a*g_{\alpha} = a^{\alpha}

Can someone explain why this is the case please?

A is a group and gαg_{\alpha} are elements of the extension, G, of A by B such that gα∗A→αg_{\alpha}*A\rightarrow \alpha is the isomorphism between G/A and B


I'm not a group theorist, but until one turns up...

We have the short exact sequence

1→A→G→B→1 \displaystyle 1 \rightarrow A \rightarrow G \rightarrow B \rightarrow 1

I think what is happening here is that gαg_{\alpha} is an element of G such that the coset gα∗Ag_{\alpha}*A is taken to α\alpha in B under the canonical isomorphism. (I've not seen that notation for cosets before, so I may be completely off wack here).gα g_{\alpha} then acts by inner automorphism on A and it is an automorphism because A is a normal subgroup of G.
(edited 8 years ago)
Reply 4
Avatar for Zacken
Zacken
22
It's been moved now. :smile:

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