by alterom | 2025-10-05 16:31:49+00:00
Eh, every industry has its own weird jargon.

Don't ask me what's normal about normal subgroups¹, for example (or what's weird about other ones, especially given that it's the normal ones that are special in some way, and the "abnormal" ones didn't even get a name).

Or why mathematicians came up with other, alternatively stupid ways to refer to them, when saying kernel subgroups (or simply kernels) would communicate the idea exactly (normal subgroups are simply kernels of morphisms, i.e. the subgroups that could be sent to identity by a map that preserves group relations).

Better yet, call them divisor subgroups (or just divisors), because the reason we care about them is that we can use them to divide a group into equally-sized pieces which also form a group.

Which we naturally call the quotient group.

So, to be clear:

* You have M=12 apples. You put them on the table in rows of N=3 apples, and get exactly 4 full rows of 3 apples each, which line up with each other:

••• ••• ••• •••

Math speak: you divide M=12 by N=3, its divisor, to get M/N=4, the quotient.

* You have a group G of remainders mod 12 . You pick three of them, H = {0, 4, 8}, and write them down in a row.

If you add 1 to each number in H, you get another, new, row of 3 remainders: 1, 5, 9. You write that down; same thing works for 2 and 3.

With modular arithnetic, adding 4 or 8 just shuffles numbers in any row that you wrote down. So adding 5 is the same as adding 1; you get only 4 rows that don't overlap.

Math speak: H is a subgroup, and H acts on each row by a shuffle. The rows correspond to remainders modulo 4.

0 4 8 = H + 0 (= H + 4 = H + 8) 1 5 9 = H + 1 (= H + 5 = H + 9) 2 6 10 = H + 2 (= H + 6 = H + 10) 3 7 11 = H + 3 (= H + 7 = H + 11)

Addition mod 12 preserves these rows too.

For example, if you pick any number from row 1 and add any number from row 2, you always get a number from row 3.

Adding two numbers from row 3 yields a number from row 2. We can write this like this:

[1] + [2] = [3] [3] + [3] = [2] [3] + [2] = [1] ...

(Here, [2] stands for "row which has 2 in it", i.e. H + 2).

You can see how the rows not only correspond to remainders mod 4 (as a set), they are related in the same way (add like numbers mod 4):

(H + a) + (H + b) = H + (a +b) = H + ((a + b) % 4) (since H + 4 = H).

So we've divided our group G=Z_12 by the action of H=Z_3 into parts which also form a group with the same structure as Z_4 (remainders mod 4).

Math speak: by analogy with the division of numbers, we say that we get a quotient group G/H=Z_4 with elements [0], [1], [2], [3].

We could use the same naming logic to say that to get the quotient group G/H, we divide G by its divisor group H....

...hahaha, who am kidding, that makes too much sense. If you saw the word divisor, you'd think of getting a quotient right away. Come on. Can't have that.

To get the quotient group G/H, you quotient G by its normal subgroup H.

Yes, mathematicians straight up verbed the word quotient, and called the operand H something that is in absolutely no way connected to it.

To recap,

Numbers: divide a number M by its divisor N do get the quotient M/N.

Groups: quotient a group G by its normal subgroup H to get a quotient group G/H.

... I'm not going to complain about "comorbidity".

¹ https://en.wikipedia.org/wiki/Normal_subgroup