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tech / sci.math / Re: group theory - Cayley diagrams

SubjectAuthor
* group theory - Cayley diagramssobriquet
+* Re: group theory - Cayley diagramsFromTheRafters
|`* Re: group theory - Cayley diagramssobriquet
| `- Re: group theory - Cayley diagramssobriquet
+- Re: group theory - Cayley diagramsMina Arvuti lähedal
`* Re: group theory - Cayley diagramsBasil Jet
 `- Re: group theory - Cayley diagramssobriquet

1
group theory - Cayley diagrams

<b746204f-f19e-4c5c-88c5-6349097fb6c0n@googlegroups.com>

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Subject: group theory - Cayley diagrams
From: dohduh...@yahoo.com (sobriquet)
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 by: sobriquet - Sat, 29 May 2021 18:59 UTC

Hi.
Lately I'm exploring this nice visual group theory course a bit:
https://www.youtube.com/watch?v=UwTQdOop-nU&list=PLwV-9DG53NDxU337smpTwm6sef4x-SCLv

I'm wondering if someone can help to point out something
obvious I must be overlooking when comparing the Cayley diagrams
of some finite groups.
The ones with double cyclic concentric rings come in two
variations.
Sometimes the arrows run in opposite directions along the
concentric circles and sometimes the arrows run in the same
direction.

For groups of order 6, 10, 14, ...
the arrows only run in opposite directions, but for groups of
order 8,12,16,... the arrows run in both opposite and equal directions.

https://i.imgur.com/fIiFrqI.png

https://nathancarter.github.io/group-explorer/GroupExplorer.html

Is there an obvious reason that arrows can't run in same direction for
groups of order 6,10,14,... ?

Kind regards and thanks in advance for any suggestions, Niek

Re: group theory - Cayley diagrams

<s8u8mh$j3k$1@dont-email.me>

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https://www.novabbs.com/tech/article-flat.php?id=60813&group=sci.math#60813

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From: erra...@nomail.afraid.org (FromTheRafters)
Newsgroups: sci.math
Subject: Re: group theory - Cayley diagrams
Date: Sat, 29 May 2021 16:37:37 -0400
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 by: FromTheRafters - Sat, 29 May 2021 20:37 UTC

sobriquet was thinking very hard :
> Hi.
> Lately I'm exploring this nice visual group theory course a bit:
> https://www.youtube.com/watch?v=UwTQdOop-nU&list=PLwV-9DG53NDxU337smpTwm6sef4x-SCLv
>
> I'm wondering if someone can help to point out something
> obvious I must be overlooking when comparing the Cayley diagrams
> of some finite groups.
> The ones with double cyclic concentric rings come in two
> variations.
> Sometimes the arrows run in opposite directions along the
> concentric circles and sometimes the arrows run in the same
> direction.
>
> For groups of order 6, 10, 14, ...
> the arrows only run in opposite directions, but for groups of
> order 8,12,16,... the arrows run in both opposite and equal directions.
>
> https://i.imgur.com/fIiFrqI.png
>
> https://nathancarter.github.io/group-explorer/GroupExplorer.html
>
> Is there an obvious reason that arrows can't run in same direction for
> groups of order 6,10,14,... ?
>
> Kind regards and thanks in advance for any suggestions, Niek

Might be due to the number of pairs being odd as opposed to even. By
pairs I mean by in that diagram think of inner and outer nodes being
spokes of a wheel. The number of 'nodes' may be even for some allowing
the opposite arrows but the 6,10,14 can be viewed as 3,5,7 as far as
spokes go.

Re: group theory - Cayley diagrams

<d3200eb8-1346-4a93-b71a-0805bab29319n@googlegroups.com>

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Subject: Re: group theory - Cayley diagrams
From: dohduh...@yahoo.com (sobriquet)
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 by: sobriquet - Sat, 29 May 2021 21:29 UTC

On Saturday, May 29, 2021 at 10:37:47 PM UTC+2, FromTheRafters wrote:
> sobriquet was thinking very hard :
> > Hi.
> > Lately I'm exploring this nice visual group theory course a bit:
> > https://www.youtube.com/watch?v=UwTQdOop-nU&list=PLwV-9DG53NDxU337smpTwm6sef4x-SCLv
> >
> > I'm wondering if someone can help to point out something
> > obvious I must be overlooking when comparing the Cayley diagrams
> > of some finite groups.
> > The ones with double cyclic concentric rings come in two
> > variations.
> > Sometimes the arrows run in opposite directions along the
> > concentric circles and sometimes the arrows run in the same
> > direction.
> >
> > For groups of order 6, 10, 14, ...
> > the arrows only run in opposite directions, but for groups of
> > order 8,12,16,... the arrows run in both opposite and equal directions.
> >
> > https://i.imgur.com/fIiFrqI.png
> >
> > https://nathancarter.github.io/group-explorer/GroupExplorer.html
> >
> > Is there an obvious reason that arrows can't run in same direction for
> > groups of order 6,10,14,... ?
> >
> > Kind regards and thanks in advance for any suggestions, Niek
> Might be due to the number of pairs being odd as opposed to even. By
> pairs I mean by in that diagram think of inner and outer nodes being
> spokes of a wheel. The number of 'nodes' may be even for some allowing
> the opposite arrows but the 6,10,14 can be viewed as 3,5,7 as far as
> spokes go.

Ah, yes. It seems related to this image showing axes of reflection.
Only if there is an even number of segments along the concentric circles,
there is an axis of symmetry through the middle of segments on opposite
sides of the concentric circle.

https://i.imgur.com/9HTGthM.png

Re: group theory - Cayley diagrams

<dc037c79-786b-4012-922f-2b799c6947aan@googlegroups.com>

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Subject: Re: group theory - Cayley diagrams
From: ya12...@mail.com (Mina Arvuti lähedal)
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 by: Mina Arvuti lähedal - Sun, 30 May 2021 13:03 UTC

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Click here to read the complete article

Re: group theory - Cayley diagrams

<7e0e4b08-f7e4-427c-9cad-1f3caf585been@googlegroups.com>

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Subject: Re: group theory - Cayley diagrams
From: dohduh...@yahoo.com (sobriquet)
Injection-Date: Tue, 01 Jun 2021 21:41:06 +0000
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 by: sobriquet - Tue, 1 Jun 2021 21:41 UTC

On Saturday, May 29, 2021 at 11:30:01 PM UTC+2, sobriquet wrote:
> On Saturday, May 29, 2021 at 10:37:47 PM UTC+2, FromTheRafters wrote:
> > sobriquet was thinking very hard :
> > > Hi.
> > > Lately I'm exploring this nice visual group theory course a bit:
> > > https://www.youtube.com/watch?v=UwTQdOop-nU&list=PLwV-9DG53NDxU337smpTwm6sef4x-SCLv
> > >
> > > I'm wondering if someone can help to point out something
> > > obvious I must be overlooking when comparing the Cayley diagrams
> > > of some finite groups.
> > > The ones with double cyclic concentric rings come in two
> > > variations.
> > > Sometimes the arrows run in opposite directions along the
> > > concentric circles and sometimes the arrows run in the same
> > > direction.
> > >
> > > For groups of order 6, 10, 14, ...
> > > the arrows only run in opposite directions, but for groups of
> > > order 8,12,16,... the arrows run in both opposite and equal directions.
> > >
> > > https://i.imgur.com/fIiFrqI.png
> > >
> > > https://nathancarter.github.io/group-explorer/GroupExplorer.html
> > >
> > > Is there an obvious reason that arrows can't run in same direction for
> > > groups of order 6,10,14,... ?
> > >
> > > Kind regards and thanks in advance for any suggestions, Niek
> > Might be due to the number of pairs being odd as opposed to even. By
> > pairs I mean by in that diagram think of inner and outer nodes being
> > spokes of a wheel. The number of 'nodes' may be even for some allowing
> > the opposite arrows but the 6,10,14 can be viewed as 3,5,7 as far as
> > spokes go.
> Ah, yes. It seems related to this image showing axes of reflection.
> Only if there is an even number of segments along the concentric circles,
> there is an axis of symmetry through the middle of segments on opposite
> sides of the concentric circle.
>
> https://i.imgur.com/9HTGthM.png

Oh wait, I see I'm mistaken about my assumption that the arrows can't go in
the same direction for groups of order 6,10,14,...

It's just that they happen to be isomorphic to other finite groups, so they don't get
represented separately in the list of finite groups.

https://i.imgur.com/zMwO29k.png

Re: group theory - Cayley diagrams

<s9edpe$mis$1@dont-email.me>

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From: bas...@spamspamspam.com (Basil Jet)
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Subject: Re: group theory - Cayley diagrams
Date: Sat, 5 Jun 2021 00:42:37 +0100
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 by: Basil Jet - Fri, 4 Jun 2021 23:42 UTC

On 29/05/2021 19:59, sobriquet wrote:
> Hi.
> Lately I'm exploring this nice visual group theory course a bit:
> https://www.youtube.com/watch?v=UwTQdOop-nU&list=PLwV-9DG53NDxU337smpTwm6sef4x-SCLv

I'm working my way through it, there seems to be a lot of mistakes in
it. The assertion in video 1.2 that every Rubik position has 6 moves
that take you nearer to solved and six that take you away is ridiculous.
If the cube is solved, all twelve moves take you away, and if you are in
the super-flip with four spot, all twelve moves take you towards solved.

--
Basil Jet recently enjoyed listening to
Mayday Signals - Swell Maps

Re: group theory - Cayley diagrams

<536fb3d6-1aef-4a48-99de-f15a87681fbbn@googlegroups.com>

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Subject: Re: group theory - Cayley diagrams
From: dohduh...@yahoo.com (sobriquet)
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 by: sobriquet - Sat, 5 Jun 2021 00:41 UTC

On Saturday, June 5, 2021 at 1:42:52 AM UTC+2, Basil Jet wrote:
> On 29/05/2021 19:59, sobriquet wrote:
> > Hi.
> > Lately I'm exploring this nice visual group theory course a bit:
> > https://www.youtube.com/watch?v=UwTQdOop-nU&list=PLwV-9DG53NDxU337smpTwm6sef4x-SCLv
> I'm working my way through it, there seems to be a lot of mistakes in
> it. The assertion in video 1.2 that every Rubik position has 6 moves
> that take you nearer to solved and six that take you away is ridiculous.
> If the cube is solved, all twelve moves take you away, and if you are in
> the super-flip with four spot, all twelve moves take you towards solved.
>
>
> --
> Basil Jet recently enjoyed listening to
> Mayday Signals - Swell Maps

Yeah, I guess many of these mistakes are pointed out in the comments
on youtube.
I've also found the book "Visual Group Theory by Nathan C. Carter"
as a pdf via libgen and that makes a nice companion to the course to explore
some additional background info to clarify some aspects that aren't covered
extensively in the videos.

The videos by Bill Shillito are also pretty good, though they don't go into more
detail about the specific connection between Galois groups and how they are
useful to understand why there can't be any closed expression to characterize the
roots of polynomials of degree 5 and higher similar to the ones for polynomials
of lower degree.
https://www.youtube.com/watch?v=WwndchnEDS4

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