My identity in high school

Question

My first semester—no, my entire freshman year—in high school, I was all alone.

As I started my sophomore year, I finally made a friend. Two, actually. They weren't in the same class as me, but for the rest of my high school life, they and their classmates basically defined my whole friend group.

As sophomore year drew to a close, I made yet more friends, from another class. There were finally some dynamics between us.

Cue the start of junior year. More friends from yet another class, yet I feel things won't be so complicated anymore. (This semester we were big into D&D, but we had to substitute d12's for d20's).

At some point during my sixth semester, we looked at ourselves in the mirror and realized we were quite exceptional.

Start of senior year: Exponent? Blaze it!

How many friends did I have by graduation, and across how many different classes?

Answer 1

Apologies for yet another answer, but I really think I finally got it, as it fits all clues and your bonus hints!

My first semester—no, my entire freshman year—in high school, I was all alone.

In semester $n$ you are the identity in $A_n$, the alternating group of even permutations of $n$ symbols. $A_1$ and $A_2$ don't have any other elements.

As I started my sophomore year, I finally made a friend. Two, actually.

Your new friends are the other two elements of $A_3$

They weren't in the same class as me, but for the rest of my high school life, they and their classmates basically defined my whole friend group.

They are not in your conjugacy class, but they and their classmates, together all 3-cycles (there are one or two conjugacy classes of 3-cycles, these two represent both classes), generate your whole friend group, namely $A_n$, which is generated by its 3-cycles.

As sophomore year drew to a close, I made yet more friends, from another class. There were finally some dynamics between us.

Next semester, more friends: $A_4$. Some from another class, namely that of the products of disjoint transpositions. Not sure about the dynamics.

Cue the start of junior year. More friends from yet another class, yet I feel things won't be so complicated anymore. (This semester we were big into D&D, but we had to substitute d12's for d20's).

Not so complicated anymore, namely simple: $A_n$ for $n \ge 5$ is simple. The new class(es) are the two classes of 5-cycles. $A_5$ is the symmetry group both of d12 and of d20.

At some point during my sixth semester, we looked at ourselves in the mirror and realized we were quite exceptional.

In the mirror you saw all your symmetries, those of $A_6$, which are exceptional: all other alternating groups have an outer automorphism group of (1 or) 2 elements, but that of $A_6$ is the Klein 4-group.

Start of senior year: Exponent? Blaze it!

The exponent of $A_7$ is 420. This is slang for smoking marihuana.

How many friends did I have by graduation, and across how many different classes?

You have $8!/2 - 1 = 20159$ friends, across 14 conjugacy classes, including your own.

Answer 2

At first I thought that you might be

finite simple groups (you a cyclic group of prime order, learned about in your first year, then maybe $A_5$, from the class of alternating group and a group of Lie type (dynamics), etc). Could also have been just sporadic simple groups, although those don't come up during your freshman year.

but there were too many clues that I couldn't explain.

Then I thought you could be

elementary particles.

There are still many parts that I doubt about, but let me try, first the global story:

I think you are some kind of a fermion, probably an electron, which may be the only elementary particle you study during your freshman year. Your first friends in the second year were some more fermions, but by the end of the sophomore year you made friends that gave rise to some dynamics: those were some gauge bosons, like the photon, and maybe weak gauge bosons: in quantum field theory, fermions interact through the exchange of bosons, so without bosons there are no dynamics. The classmates of your first 2 friends defined your whole group of friends, even though several are from other classes. That is because the gauge bosons arise purely from the fermion dynamics when you impose local gauge invariance.

In junior year, more friends, but a simplification anyhow. Could that be the addition of quarks, which revealed that hadrons and mesons were not actually elementary but had substructure, and instead of having scores of hadrons and mesons, you could do with a handful of quarks?

Looking in the mirror, you might see your antiparticle. It might also be you superpartner. Not sure in which sense your group of friends is exceptional though.

Classes could be fermion generations or particle types (fermions, vector bosons, scalar bosons) or also fermion types (leptons or quarks). To make it work in this story, different ones of these could serve.

To fill in the details, there are different stories possible, none of them much more convincing than any other, so I'm not so sure.

Some clues that I have no idea how to interpret:

Not sure about the d12 and d20, they seem to be dice shaped like a dodecahedron or an icosahedron, but since both have the same symmetry group, it seems unnecessary to change one for the other in the context of elementary particles.

No idea what happened in senior year...

Let me make a wild guess, I hope that I'm at least somewhat on the right track:

The classes are three generations of leptons, then quarks (let's restrict them to one class) and bosons: 5 classes.

You are the electron, your first friends where the muon and the tau-lepton, then the photon, two quarks and your new friends from senior year, but let's not count them because I don't know how to interpret the clue. You would have at least 5 friends, but possibly also quarks of different generations and additional gauge bosons. The Higgs doesn't seem to enter, and neutrino's not explicitly. Most of these are different from their antiparticle, so if you befriended those as well, there are many more. The quarks also have a color charge, so if we count those as different friends, there would be many more again.

Answer 3

Ok, with your hint, I'll take another stab at it. I think you and your friends are

elements of some group satisfying some property.

You were in a

conjugacy class of one element.

The friends you made, were in a different

conjugacy class.

They, together with the others in their class(es)

generate the whole group.

There are at least two more

conjugacy classes.

You and your friends are exceptional, as you can see from looking in the mirror. This could mean that

all of you are elements that are their own inverse, or that your conjugacy classes are closed under taking inverses.

Swapping d12 for d20:

Maybe there are no more friends for you in $D_{12}$, the dihedral group of 24 (or of 12) elements (both notations are seen), but you can repeat the preceding in $D_{20}$, in which case you still have 2 more conjugacy classes.

So, for the full story:

I am the identity in the dihedral group $D_{20}$ of order 20. The conjugacy classes of my first two friends generate the whole group, so they could be e.g. $s$ and $rs$, where $r,s$ are such that $D_{20} = \langle r,s\,|\,r^{10},s^2,srsr\rangle$. Other friends from other conjugacy classes. We are exceptional in the mirror: the inverse of each element is in the same conjugacy class, i.e. mirror images are classmates. I am not sure if "blaze it!" means something really good, but it certainly sounds as if it does, just like having a 10, the exponent of $D_{20}$!

All in all

20 friends (including myself) over 8 classes.