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I want ideas for pictorial representation of groups which can help one understand the different group theorems.

Here are some examples of the type of thing I am looking for. In this video by socratica at 6:51, the following picture comes up on screen to show that group set can be spanned by premultiplying the coset by a group element for proving lagranges theorem.

enter image description here

Of course, the above representation is not perfect but it's a direction in how one could think about these things.

Here is another example which I saw for the orbit stabilizer theorem from this post:

enter image description here

The visualization is so simple and captures the big picture of the proof so easily. Also the other attached picture which the author uses to prove that the orbit of any element $x$ in the orbit set of another element $y$ has equal length to orbit set of $y$.

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3 Answers 3

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The book Visual Group Theory by Nathan Carter seems to be a rich source of the kind of materials that you are looking for.

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  • $\begingroup$ See also Macauley's video series based in part on Carter's book. $\endgroup$
    – J W
    Mar 12 at 17:04
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John Jones has a great visualization tool for a nice selection of finite group tables. It allows you to do things like select a subgroup, and see the group table colored according to the cosets. This lets you see "at a glance" whether a given subgroup is normal or not.

https://hobbes.la.asu.edu/groups/groups.html

normal subgroup

non-normal subgroup

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The Cayley Graph is visually appealing. Here is the dihedral group $D_4$ on two generators $a$ and $b$:

D4
(Wikipedia image)

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