The seminar is now complete! A brief log of what we covered is below—notes for the seminar can be found on GitHub:
Seminar Notes: An Introduction to Permutation Groups
Please email Joshua Wiscons [joshua.wiscons@csus.edu] if you have any questions or notice any errors.
About the Series
This is a multi-part series on permutation groups. The target audience is undergraduate (and beyond) math-interested folk with exposure to linear algebra. Familiarity with permutations and modular arithmetic (as one would see in Math 110A) will also be useful, but not necessary. It is a joint offering of the Algebra, Number Theory, and Combinatorics (ANTC) seminar and the Algebra & Logic Lab at Sacramento State. As with the 2021 seminar series, the main goals of the series are for participants to have fun, learn something interesting they wouldn’t see in a regular class, and get excited to study more math. Perhaps the seminar will be a starting point for an independent study or research project.
Notes for the series can be found at the link below. They are actively being developed/revised/updated as the seminar progresses. Please email Joshua Wiscons [joshua.wiscons@csus.edu] if you have any questions or notice any errors.
Seminar Notes: Topics in Permutation Groups
The series is made possible in part by support from the National Science Foundation under Grant No. DMS-1954127. Any opinions, findings, and conclusions or recommendations expressed on this website are those of Joshua Wiscons and do not necessarily reflect the views of the National Science Foundation.
Tentative Topics
- Introduction to permutation groups: notation and terminology.
- Examples:
- symmetric and alternating groups
- groups acting on graphs
- linear, affine, and projective groups
- some Mathieu groups (via actions similar to those that can be performed on a Dogic)
- Classification of permutation groups with a high degree of transitivity
- Relational complexity
Tentative Schedule
- Meeting 1: February 25 from 3:00–4:00
- Meeting 2: March 04 from 3:00–4:00
- Meeting 3: March 18 from 3:00–4:00
- Meeting 4: April 01 from 3:00–4:00
- Meeting 5: April 15 from 3:00–4:00
- Meeting 6:
April 22May 06 from 3:00–4:00
Meeting 1
February 25 from 3:00–4:00
In our first meeting, we’ll discuss our goals for the series and explore some background on permutations, permutation groups, and the symmetric group.
The Zoom link and password for our meeting will be distributed via email. Please email joshua.wiscons@csus.edu if you have questions or would like to be added to the email list.
Topics
- Welcome, introductions, framing, and goals
- Cycle notation for permutations
- First examples: the symmetric group and the symmetry group of a regular tetrahedron
Preparing for the meeting
There is nothing you need to do to prepare. If you want to preview what we’ll discuss, you can follow the links in the list of topics above. And if you haven’t already, you may just want to watch this video. (Thanks again to BC for the suggestion.)
Summary from Meeting 1
After introductions, we spent our time working out the symmetries of a regular tetrahedron. Along the way, we introduced/reviewed and bit of terminology about permutations, including two-line and cycle notation for permutations. Looking forward to seeing everyone next week!
Meeting 2
March 04 from 3:00–4:00
The Zoom link and password are the same as for our first meeting. Please email joshua.wiscons@csus.edu if you have questions.
Topics
- Permutation groups: more notation and terminology
- Groups acting on sets
- Examples: the alternating group and the symmetry group of the cube
Preparing for the meeting
Again there is nothing you need to do to prepare. But here’s a video discussing the mathematics (of permutations) behind the mind switching problem that appears (and is solved) in the Prisoner of Brenda episode of Futurama. (The episode was written by Ken Keeler who has a PhD in Applied Math.)
Summary from Meeting 2
We started with the proper defintion of a permutation group. After that, we talked about even and odd permutations and introduced the alternating group, connecting back to the rotational symmetries of the tetrahedron. Finally, we defined group actions and started investigating the symmetries of a cube. Our initial point of view (as with the tetrahedron) is as permutations of the vertices, but next time we see the value in considering a related action on the “diagonals” of the cube.
Meeting 3
March 18 from 3:00–4:00
The Zoom link and password are the same as for our first meeting. Please email joshua.wiscons@csus.edu if you have questions.
Topics
Summary from Meeting 3
We continued looking at symmetries of the cube. This time we studied how they permute the diagonals (in addition to the vertices), and this led to a very nice description of the rotational symmetries. We also took a look at the symmetries of the icosahedron but satisfied ourselves with simply counting the number of rotational symmetries. We then started to formalize our approach to counting these symmetries by introducing the notions of orbits and stabilizers. Next time we’ll look at an important numerical relationship between orbits and stabilizers that was underlying our approach to counting the rotational symmetries of the icosahedron.
Meeting 4
April 01 from 3:00–4:00
The Zoom link and password are the same as for our first meeting. Please email joshua.wiscons@csus.edu if you have questions.
Topics
Summary from Meeting 4
We discussed the orbit-stabiliter theorem and revisited our counting of the rotational symmetries of the icosahedron through the lens of the theorem. We then transitioned into a new topic: k-transitivity. After giving the definition, we reviewed a couple of our earlier examples (about symmetries of the tetrahedron and the cube) to highlight the degree of transitivity of those actions.
Meeting 5
April 15 from 3:00–4:00
The Zoom link and password are the same as for our first meeting. Please email joshua.wiscons@csus.edu if you have questions.
Topics
- k-transitive actions
- Affine and projective groups
- Some Mathieu groups (via actions similar to those that can be performed on a Dogic)
- Classification of permutation groups with a high degree of transitivity
Summary from Meeting 5
The focus of our meeting was on $k$-transitivity. We began by working out that the natural action of $\operatorname{Sym}(5)$ is $5$-transitive and then remarking that this generalizes to show $\operatorname{Sym}(n)$ is $n$-transitive. We then met the group $\operatorname{AGL}_1(\mathbb{R})$, and showed that its action on $\mathbb{R}$ is $2$- but not $3$-transitive (and the same is true for $\operatorname{AGL}_1(F)$ for almost every field $F$). After that, we met another new group: $\operatorname{PGL}_2(\mathbb{R})$. This group turns out to act $3$- but not $4$-transitively on the projective line. Next, in our quest to find more groups with a high degree of transitivity (other than $Sym(n)$ and $Alt(n)$), we met the Mathieu group $M_{12}$ as the collection of operations on the vertices of the icosahedron generated by certain “twist-untwist” moves (similar to what can be done on a Dogic). We remarked that $M_{12}$ acts $5$-transitively, and then concluded the meeting by presenting a deep theorem that lists all finite permutation groups with a $4$-transitive action: $\operatorname{Sym}(n)$, $\operatorname{Alt}(n)$, and four Mathieu groups ($M_{11}$,$M_{12}$,$M_{22}$, and $M_{23}$).
Meeting 6
May 06 from 3:00–4:00
The Zoom link and password are the same as for our first meeting. Please email joshua.wiscons@csus.edu if you have questions.
Topics
- Relational complexity of permutation groups: examples, open problems, and current research
Summary from Meeting 6
For our last meeting, Meagan presented an introduction to the relatively new topic of relational complexity. After discussing (and illustrating) the definition, Meagan took us through a couple of examples. She first talked us through determining the complexity of $\operatorname{Sym}(n)$ and then worked out the complexity of $\operatorname{AGL}_1(F)$. The talk finished by introducing the semilinear group $\operatorname{A\Gamma{}L}_1(F)$, and discussing current research to try to determine the complexity of this group. Many thanks to Meagan for a great presentation, and many thanks to you all for a fun semester!
