My feeling is that the division/factorization issue represents some sort of basic awareness of numbers as "entities", rather than just digit strings.

My own awareness of numbers is synaesthetic, they seem to have shapes which help me see how they "break apart" into their factors. Sometimes the shapes reflect the sub-factors, or the square or triangular "status" (some numbers can take more than one shape), but they're not always something I could draw or shape from clay.

Plurals: I just had the same issue with pluralizing the phrase "gentleman's C" someplace else. I think the standard rules of English don't quite cover using letters as symbols, but the real point is readability, so I'd do the same as you did.

The apostrophe use is fine, since you're ducking out of the English Language for a moment to describe something that it doesn't really cover, it's OK to defy convention.
I reckon that the whole "numbers as entities" thing is a bit of a problem in the way that maths is taught but I for one, couldn't think of an alternative way of teaching it to little ones.

Two things. First, regarding the apostrophe, I was specifically taught that one of the proper uses of the apostrophe was pluralizing numbers and symbols. So in fact (at least according to *my* junior high grammar text) it would have been wrong *not* to include it. Secondly, regarding division. It seems to me that this is an interesting parallel to the fact that understanding the construction of quotient groups (or rings or whatever) is a sort of "litmus test" for understanding abstract algebra... Perhaps there is something fundamental going on here on a cognitive psychology level? Intuitively, I agree with David that the problem probably lies with the ability to see numbers as things-in-themselves. (And,in the abstract algebra case, with seeing groups, etc. as things-in-themselves.)

Interesting that you would bring up quotient groups in this context. I never really made that connection myself, but that probably was one of the abstractions that brought me to the conclusion that I wasn't cut out to be a mathematician. I can follow the definitions of quotient groups, but I don't think I ever got an intuition for them. That, and normal subgroups. I was never able to find an example of a normal vs. non-normal subgroup that really made me understand why we care about normal ones. A reference to a good post online somewhere about it would be appreciated.

for quotient groups, begin with the integers-mod-n.

for normal subgroups ... well, they're the ones that allow the formation of quotient groups!

"D's" looks right to me, too.

oh, and. minimal counterexample.

we call of subgroup H of G *normal in G* if the collection of the "left cosets"

{xH | x \in G}

coincides with the collection of "right cosets"

{Hx | x \in G}.

when H is normal, we can make a group out of the set
of its (left) cosets via (xH)*(yH) = (xy)H.

let G denote the symmetric group on three elements:

G = {(), (12), (23), (13), (123), (132)}.

consider the subgroup

B = {(), (12)}.

the left cosets of B are then:

B itself,

(13)B = {(13), (123)} (= (123)B), and

(23)B = {(23), (132)} (= (132)B).

but these do *not* form a group in the manner we considered a moment ago (as this would entail

(13)B*(12)B = (123)B \not= B

but at the same time

(123)B*(132)B = B

see how easy that was?

vlorbik,

when i read the definitions for normal subgroups, i can follow them when i take excruciating care to calculate all the cosets (like you showed). and i know that every normal subgroup generates quotient groups with special properties involving isomorphisms between the subgroup and the kernel, and...something...i can't remember.

but i've never seen a example that makes me say "oh, THAT's why a normal subgroup is so much cooler."

not that it's your responsibility to find one--i think it's just that i haven't learned to think about the abstractions well enough.

An example that (perhaps) shows why normal subgroups are so much cooler.

Consider the set of n by n matrices with entries in the real numbers and determinant one. This set (which we call SL_n) forms a group with the operation of matrix multiplication. Now, SL_n has lots and lots of subgroups. For instance, we could take any element A of SL_n and look at the set A,A^2,A^3, et cetera. Or we could look at all the matrices in SL_n that have rational entries. (It's pretty clear that these form a subgroup.) And so on and so forth. We could go on defining subgroups of SL_n all day, if we felt like it. But what about normal subgroups? These there are very few of! In fact, if n is odd, there is only the trivial subgroup, and if n is even, there is only the trivial subgroup and the one generated by -I (where I is the identity matrix). (The general fact, if we allowed, say, complex numbers, is that the only normal subgroups are the ones that are generated by scalar multiples of I.) Intuitively, what is going on is that conjugation (in any matrix group) amounts to a change of basis, so in order to be normal, a subgroup must be invariant under (certain) changes of basis. Clearly, scalar multiples of the identity have this property; intuitively it makes sense that nothing else does. (As usual, proving this is another matter.)

For another example, take the case of the symmetric group on a set X. In this case, conjugation is precisely equivalent to a "renaming" of the elements of X. So in order to be normal, a subgroup must be equivalent to itself under all relabelings of the underlying set. In the case X = {1,2,3} discussed above, the subgroup {(123),(132),()} is normal because we can clearly relabel X however we like and not change the subgroup. The subgroup {(12),()} is not normal because if we swap 1 and 3, it becomes a different subgroup, {(32),()}. Aside: it can be shown that ,as long as |X| > 4, the symmetric group on X has only one normal subgroup!

Two final notes. First, I think a key to understanding quotient groups (at least for me) is making the leap from the analogy

group elements == numbers

to the analogy

groups == numbers.

Second, it helps, when trying to understand the concept of a normal subgroup, to keep in mind that an alternative definition of "normal" is that a normal subgroup is the kernel of some homomorphism. This is a much more natural definintion than the one about being fixed under conjugation, and it also highlights the analogy between normal subgroups in group theory and ideals in ring theory. Anyway, I've gone on long enough now... hope some of it helps :-)

For some reason A-Level maths includes practically no group theory, thus I understand roughly none of the comments on it. Do they teach group theory in US high schools or do you lot just know it from uni or something?

paul carpenter: right, from uni. i first saw this stuff senior year.

cat lover: very nice exposition! conjugation as relabeling ... why didn't *i* mention that?

polymath: i still maintain that Z/nZ is the killer example you seek. the point is that we put a group structure on the cosets: the very *elements* of, say, Z/2Z are

[0]:={...,-2, 0, 2, 4,...}

(the identity element) and

[1]:= {...,-3,-1,1,3,...}

(the other coset of [0] in Z). that stuff you don't remember is that the kernel (the normal subgroup) becomes the identity element in a new group etcetera. actually, i'm just guessing ... this might have been clear to you already ... far be it from me to ask you to give this matter any more thought than you feel it deserves ....

Perhaps part of the reason that normalness is such a non-obvious property is that most "obvious" groups (the ones we understand the best) are all abelian, so there are no non-normal subgroups.

Normal subgroups are closed under a sort of "conjugacy":

gh(g^-1) is in H for all g in G

(for all h in H, where H is a normal subgroup of G).

(In fact, thinking about it, this is precisely why G/H can be formed).

This general idea - that you can do APPLE (g) to something, then do ORANGE (h), then do APPLE^-1, and it's the same effect as if you just did an ORANGE - occurs over and over again in many diverse areas of maths.

e.g. using Fourier transforms, you can transform a problem to phase space, do a simplified calculation, then transform back - and it works just the same as if you'd done the (more complicated) calculation in the problem space.

another e.g. - the construct PA(P^-1), which occurs over and over again in linear algebra.

In a very general way, I guess this useful type of structure is part of the underlying motivation for considering normal subgroups.

SL

thanks for that explanation. that's just the kind of thing i was looking for.

i can see now how imposing a group structure on a situation (and finding normal subgroups) could help in certain kinds of calculations or analysis, even though i still don't quite have an intuition for what they are.

much appreciated.

I'm appropriating the "plane that won't crash and kill people" schtick. Thank you.

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