In the previous post Pattern recognition and me, I wrote about how much I enjoyed sudden flashes of understanding that were caused by my recognizing a pattern (or learning about a pattern). I have had several such, shall we say, Thrills in learning about math and doing research in math. This post is about a very early thrill I had when I first started studying abstract algebra. As is my wont, I will make various pronouncements about what these mean for teaching and understanding math.

Cosets

Early in any undergraduate course involving group theory, you learn about cosets.

Basic facts about cosets

1. Every subgroup of a group generates a set of left cosets and a set of right cosets.
2. If $H$ is a subgroup of $G$ and $a$ and $b$ are elements of $G$, then $a$ and $b$ are in the same left coset of $H$ if and only if $a^{-1}b\in H$. They are in the same right coset of $H$ if and only if $ab^{-1}\in H$.
3. Alternative definition: $a$ and $b$ are in the same left coset of $H$ if $a=bh$ for some $h\in H$ and are in the same right coset of $H$ if $a=hb$ for some $h\in H$
4. One of the (left or right) cosets of $H$ is $H$ itself.
5. The relations
   $a\underset{L}\sim b$ if and only if $a^{-1}b\in H$

   and

   $a\underset{R}\sim b$ if and only if $ab^{-1}\in H$

   are equivalence relations.

6. It follows from (5) that each of the set of left cosets of $H$ and the set of right cosets of $H$ is a partition of $G$.
7. By definition, $H$ is a normal subgroup of $G$ if the two sets of cosets coincide.
8. The index of a subgroup in a group is the cardinal number of (left or right) cosets the subgroup has.

Elementary proofs in group theory

In the course, you will be asked to prove some of the interrelationships between (2) through (5) using just the definitions of group and subgroup. The teacher assigns these exercises to train the students in the elementary algebra of elements of groups.

Examples:

1. If $a=bh$ for some $h\in H$, then $b=ah’$ for some $h’\in H$. Proof: If $a=bh$, then $ah^{-1}=(bh)h^{-1}=b(hh^{-1})=b$.
2. If $a^{-1}b\in H$, then $b=ah$ for some $h\in H$. Proof: $b=a(a^{-1}b)$.
3. The relation “$\underset{L}\sim$” is transitive. Proof: Let $a^{-1}b\in H$ and $b^{-1}c\in H$. Then $a^{-1}c=a^{-1}bb^{-1}c$ is the product of two elements of $H$ and so is in $H$.

Miscellaneous remarks about the examples

• Which exercises are used depends on what is taken as definition of coset.
• In proving Exercise 2 at the board, the instructor might write “Proof: $b=a(a^{-1}b)$” on the board and the point to the expression “$a^{-1}b$” and say, “$a^{-1}b$ is in $H$!”
• I wrote “$a^{-1}c=a^{-1}bb^{-1}c$” in Exercise 3. That will result in some brave student asking, “How on earth did you think of inserting $bb^{-1}$ like that?” The only reasonable answer is: “This is a trick that often helps in dealing with group elements, so keep it in mind.” See Rabbits.
• That expression “$a^{-1}c=a^{-1}bb^{-1}c$” doesn’t explicitly mention that it uses associativity. That, too, might cause pointing at the board.
• Pointing at the board is one thing you can do in a video presentation that you can’t do in a text. But in watching a video, it is harder to flip back to look at something done earlier. Flipping is easier to do if the video is short.
• The first sentence of the proof of Exercise 3 is, “Let $a^{-1}b\in H$ and $b^{-1}c\in H$.” This uses rewrite according to the definition. One hopes that beginning group theory students already know about rewrite according to the definition. But my experience is that there will be some who don’t automatically do it.

in beginning abstract math courses, very few teachers
tell students about rewrite according to the definition. Why not?

• An excellent exercise for the students that would require more than short algebraic calculations would be:
  • Discuss which of the two definitions of left coset embedded in (2), (3), (5) and (6) is preferable.
  • Show in detail how it is equivalent to the other definition.

A theorem

In the undergraduate course, you will almost certainly be asked to prove this theorem:

A subgroup $H$ of index $2$ of a group $G$ is normal in $G$.

Proving the theorem

In trying to prove this, a student may fiddle around with the definition of left and right coset for awhile using elementary manipulations of group elements as illustrated above. Then a lightbulb appears:

In the 1980’s or earlier a well known computer scientist wrote to me that something I had written gave him a satori. I was flattered, but I had to look up “satori”.

If the subgroup has index $2$ then there are two left cosets and two right cosets. One of the left cosets and one of the right cosets must be $H$ itself. In that case the left coset must be the complement of $H$ and so must the right coset. So those two cosets must be the same set! So the $H$ is normal in $G$.

This is one of the earlier cases of sudden pattern recognition that occurs among students of abstract math. Its main attraction for me is that suddenly after a bunch of algebraic calculations (enough to determine that the cosets form a partition) you get the fact that the left cosets are the same as the right cosets by a purely conceptual observation with no computation at all.

This proof raises a question:

Why isn’t this point immediately obvious to students?

I have to admit that it was not immediately obvious to me. However, before I thought about it much someone told me how to do it. So I was denied the Thrill of figuring this out myself. Nevertheless I thought the solution was, shall we say, cute, and so had a little thrill.

A story about how the light bulb appears

In doing exercises like those above, the student has become accustomed to using algebraic manipulation to prove things about groups. They naturally start doing such calculations to prove this theorem. They presevere for awhile…

Scenario I

Some students may be in the habit of abandoning their calculations, getting up to walk around, and trying to find other points of view.

1. They think: What else do I know besides the definitions of cosets?
2. Well, the cosets form a partition of the group.
3. So they draw a picture of two boxes for the left cosets and two boxes for the right cosets, marking one box in each as being the subgroup $H$.
4. If they have a sufficiently clear picture in their head of how a partition behaves, it dawns on them that the other two boxes have to be the same.

Remarks about Scenario I

• Not many students at the earliest level of abstract math ever take a break and walk around with the intent of having another approach come to mind. Those who do Will Go Far. Teachers should encourage this practice. I need to push this in abstractmath.org.
• In good weather, David Hilbert would stand outside at a shelf doing math or writing it up. Every once in awhile he would stop for awhile and work in his garden. The breaks no doubt helped. So did standing up, I bet. (I don’t remember where I read this.)
• This scenario would take place only if the students have a clear understanding of what a partition is. I suspect that often the first place they see the connection between equivalence relations and partitions is in a hasty introduction at the beginning of a group theory or abstract algebra course, so the understanding has not had long to sink in.

Scenario II

Some students continue to calculate…

1. They might say, suppose $a$ is not in $H$. Then it is in the other left coset, namely $aH$.
2. Now suppose $a$ is not in the “other” right coset, the one that is not $H$. But there are only two right cosets, so $a$ must be in $H$.
3. But that contradicts the first calculation I made, so the only possibility left is that $a$ is in the right coset $Ha$. So $aH\subseteq Ha$.
4. Aha! But then I can use the same argument the other way around, getting $Ha\subseteq aH$.
5. So it must be that $aH=Ha$. Aha! …indeed.

Remarks about Scenario 2

• In step (2), the student is starting a proof by contradiction. Many beginning abstract math students are not savvy enough to do this.
• Step (4) involves recognizing that an argument has a dual. Abstractmath.org does not mention dual arguments and I can’t remember emphasizing the idea to my classes. Tsk.
• Scenario 2 involves the student continuing algebraic calculations till the lightbulb strikes. The lightbulb could also occur in other places in the calculation.

References

• Cosets in Wikipedia.
• Normal subgroup in Wikipedia.
• Equivalence Relations. Article in abmath.
• Pattern recognition and me. Post in G&G.
• Pattern recognition in understanding math. Post in G&G.
• Pattern recognition. Article in abmath.
• Rabbits. Article in abmath.
• Representations and models. Article in abmath.
• Rewrite according to the definition. Article in abmath.

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