Group Theory (Automorphisms)

A permutation
is a bijection
from a set
to itself.

An automorphism
is an isomorphism
that’s a permutation
as well.

Automorphisms form
the group Aut(G).
An automorphism is like
a symmetry of symmetries.

It changes labels,
but never group relations,
nothing observable
about the group changes.

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Fix an element g in your set,
and conjugate every x,
send x to g x g inverse,
mapping the group back to itself.

Inner automorphisms,
(labels rearrange).
Inner automorphisms,
(relations never change).

As maps within the group,
they may be abundant.
As degrees of freedom,
they are redundant.

Inner automorphisms,
(labels rearrange).
Inner automorphisms,
(relations never change).

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Outer automorphisms,
are structural, in fact,
they are relabelings
no element can enact.

Out(G)
is Aut(G)
mod Inn(G).

Out(G)
is Aut(G)
mod Inn(G).

No general formula,
and Out(G) might be wild,
something of a mystery,
yet sometimes classified.

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Written by: Aitlantis Civic

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