
Visual Group Theory: What is a group? (2016) [video] - adamnemecek
https://www.youtube.com/watch?v=UwTQdOop-nU
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ginnungagap
Group theory originated in a very visual and geometric ways studying the
symmetries of various objects, before the modern definition of group was
given. Many groups studied today are still geometrical in nature: the dihedral
groups are the groups of symmetries of regular polygons, the symmetric groups
are the groups of permutations of a set with N elements and so on and so
forth.

After the notion of group was formalized (a set with an operation satisfying
some axioms) it was natural to ask oneself what was the advantage of this
apparently more general and abstract definition. Turns out (that's a theorem
of Cayley) that every group is isomorphic to a subgroup of a symmetric group
(that is a group of permutations on some set) so arguably we didn't get that
much more generality while moving from the first geometric groups to the
modern abstract ones!

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usr381
Can't pass an opportunity to link Arnold's excellent _On teaching mathematics_
([https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html](https://www.uni-
muenster.de/Physik.TP/~munsteg/arnold.html)) for those who have not read it.
As he states:

> What is a group? Algebraists teach that this is supposedly a set with two
> operations that satisfy a load of easily-forgettable axioms. This definition
> provokes a natural protest: why would any sensible person need such pairs of
> operations? "Oh, curse this maths" \- concludes the student (who, possibly,
> becomes the Minister for Science in the future).

> We get a totally different situation if we start off not with the group but
> with the concept of a transformation (a one-to-one mapping of a set onto
> itself) as it was historically. A collection of transformations of a set is
> called a group if along with any two transformations it contains the result
> of their consecutive application and an inverse transformation along with
> every transformation.

> This is all the definition there is. The so-called "axioms" are in fact just
> (obvious) properties of groups of transformations. What axiomatisators call
> "abstract groups" are just groups of transformations of various sets
> considered up to isomorphisms (which are one-to-one mappings preserving the
> operations). As Cayley proved, there are no "more abstract" groups in the
> world. So why do the algebraists keep on tormenting students with the
> abstract definition?

~~~
sorokod
> So why do the algebraists keep on tormenting students with the abstract
> definition?

In my personal experience, the ability to extract the motivation and build up
intuition is considered to be skill that the "students" have to develop for
themselves and this is part of their learning process. Those that are not
sufficiently successful will fail - and that is OK in such a system.

~~~
gowld
You'd prefer that a century of professional mathematicians of the past never
have had their careers, since their way of thinking doesn't pass your bar of
complexity?

~~~
sorokod
Not my bar, personally I think this is a crap attitude but one that I
experienced.

~~~
posterboy
It should be noted that group theory, as far as i can see, is a rather new
development and a century is not much in the evolution of didactic.

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0xFFC
This is so fantastic. Thank you so much.

I am downloading all of his channel right now to watch and I don't know how to
thank you :)

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gowld
There are very few pictures and no animations, in this "Visual" Group Theory.
It's almost all text slides.

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adamnemecek
You might be interested in the project that comes with the book that this
course is based on groupexplorer.sourceforge.net

