Re: Group Theory - clarification/motivation

In article <zLivf.192868$V7.91243@xxxxxxxxxxxxxxxxxxxxxxxxxx>,
Guy Corrigall <guy@xxxxxxxxxxxxxxxx> wrote:
>I'm unclear as to the origin/motivation for the following group theory
>terms:
>
>conjugacy - center - centralizer - annihilator - orbit - alternating -
>kernel
>
>In text after text I come across these colorful terms, I understand the
>definitions and their usage, but have never seen any history/explanation of
>their origin.
>
>For example, am I right in thinking that conjugacy originates from the
>theory of equations and conjugate roots of polynomials?

Yes. If G is a Galois group of a Galois extension K/F, then the
subgroups of G that are closed under conjugation are exactly the
subgroups that correspond to subextensions L/F (L contained in K)
which are also Galois. Conjugacy is an easy outgrow from that. Once
you have conjugacy, there is a natural map from a group into the
automorphism group of G (mapping an element to the automorphism given
by conjugating by that element); the center is the kernel of this
map. Centralizers then are natural generalizations of centers.

Kernels arise both in analogy to the linear algebra case (elements
mapped to 0), and also because they turn out to be "equivalent" to the
concept of normal subgroup (which is very important in Galois theory):
a subgroup is normal if and only if it is the kernel of a
homomorphism.

The concepts of orbit and annihilator have to do with the concept of
action: again, if you think of a group G as a Galois group (or as a
group of permutations, as was done), then you are interested in how
the objects you are acting on (the roots of the polynomial in the
Galois group case) behave under the action of the group.

>If so, would this
>explain why the inner automorphisms of a group are conjugate elements?

The inner automorphisms are NOT conjugate elements. They are
automorphisms. What is true is that the inner automorphisms
->correspond<- to conjugating ->by<- an element.n


--
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"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
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Arturo Magidin
magidin@xxxxxxxxxxxxxxxxx

.