Mathematically Bent Colin Adams, Editor

The proof is in the pudding.

Opening a copy of The Mathematical Intelligencer

you may ask yourself uneasily, ‘‘What is this

anyway—a mathematical journal, or what?’’ Or

you may ask, ‘‘Where am I?’’ Or even ‘‘Who am I?’’

This sense of disorientation is at its most acute when

you open to Colin Adams’s column.

Relax. Breathe regularly. It’s mathematical, it’s a

humor column, and it may even be harmless.

� Column editor’s address: Colin Adams,

Department of Mathematics,

Bronfman Science Center, Williams College,

Williamstown, MA 01267, USA

e-mail: Colin.C.Adams@williams.edu

Group TherapyCOLIN ADAMS

DDr. Stew: Hello, come on in. You’re Hank, right? I’dlike you to meet the group. This is Karen, and Bill,and Amanda and Sylvia. I’m Dr. Johnson, but you

can call me Dr. Stew. Is this your first experience with grouptherapy?Hank: Uh, yes, it is.Dr. Stew: Well I think you will find it very helpful. Whydon’t you take a seat over there between Karen and Bill.They’re married, but they don’t mind sitting apart.Hank: Married? You allow two people who are married inthe same group therapy session?Dr. Stew: Why not? Since they have a child, we know theycan multiply.

(Laughter from the group. Hank laughs along nervously.)Dr. Stew: Now, the basic idea of group therapy is to asso-ciate our issues and problems with concepts in grouptheory.Hank: What?Dr. Stew: You know, the properties of abstract groups.Hank: I’m confused.Dr. Stew: You are here for group therapy, aren’t you?Hank: But group therapy refers to the fact that there is agroup of us here in the room.

(Everyone laughs.)Dr. Stew: Hardly. It refers to a group. You know, a set ofelements with a multiplicative operation, inverses, etc. Youhave had Abstract Algebra, haven’t you?Hank: Yes, but …Dr. Stew: Good.Well, let’s get started. Sylvia, youwere talkinglast week about how your mother favors you over your sister.Do you think she has put a partial ordering on your wholefamily?Sylvia: No question. Often the poset structure becomesmore important than the algebraic structure.Dr. Stew: Does that bother you?Sylvia: Yes, it seems misguided. Why should a set-theoreticconstruct supersede an algebraic one? Over many years, wehave built up a sophisticated set of relations that haveallowed us to interpret our family relationships as a group.To give up all that structure seems counterproductive.Hank: Wait a minute. How sophisticated an algebraic objectcan it be? After all, your family is a finite set.Sylvia: Are you implying that finite groups can’t be inter-esting? What about the general linear group of dimension nover a finite field? What about the Weyl groups? I find yourattitude quite condescending.Hank: I’m sorry. I didn’t mean to …

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Sylvia: My family happens to be isomorphic to the quasid-ihedral group. And I am proud of that.Dr. Stew: Please everyone. Let’s try to be there for eachother. Hank, we have a rule here. ‘‘Our support shouldalways have measure 1.’’Hank: Sorry, I just …Dr. Stew: Okay, let’s turn to Karen. How are things with youthis week?Karen: Well, we had family over for the holidays, and what adisaster that was. My younger sister Emily, who was recentlydivorced, showed up with her new boyfriend Frank. Oh,was he obnoxious. He made lewd jokes, insulted mygrandmother to her face, and accused my mother of inten-tionally giving him the smallest pork chop. Then, afterdinner, my older sister Claire showed up with her new beauand it was none other than Emily’s ex-husband Craig.Everyone was stunned. Especially since we all couldn’tstand Craig when he was married to Emily.Sylvia: What happened?Karen: Amazingly enough, Frank and Craig hit it off. Theytook turns insulting the decor, the food and members of thefamily. They were awful.Dr. Stew: Well, the subgroup generated by Craig and Frankseems to be a problem.Hank: All families have subgroups like this. Sounds normalenough to me.Karen: What? Are you kidding? There is no way this sub-group is normal. Just conjugate it by Emily, and you don’tget the same subgroup.

Bill: Hank, where exactly did you take group theory?Sylvia: Are you confusing conjugate with conjugal?Hank: That wasn’t what I meant.Dr. Stew: If the subgroup generated by Frank and Craigwere normal, then the family could quotient out by thesubgroup and they would have a perfectly functioningfamily group again. But unfortunately, the subgroup is notby any means normal.Hank: I didn’t mean normal in the group-theoretic sense.You’re making this all so complicated. It seems to me itshould be simple.Bill: Well, it’s one thing to acknowledge that the subgroupgenerated by Frank and Craig is not normal, but to claimnone of the proper subgroups is normal, well that’s anothermatter.Karen: (Angry.) What makes you think that there isn’t asingle proper subgroup in my entire family that is normal?You don’t even know my family. It’s incredibly presump-tuous on your part.Hank: No, I didn’t mean …Dr. Stew: Okay, I think we had better move on. Hank,please try to be considerate of other group member’s situ-ations. Let’s do some free association. I say a word, you saywhat you think of. We’ll go around the room, starting withAmanda. Blue.Amanda: Carrot.Dr. Stew: Good. Sylvia, weasel.Sylvia: Chocolate.Dr. Stew: Good. Bill, clarinet.Bill: Horse.Dr. Stew: Good. And for Hank, wingnut.

Hank: Excuse me, but I’m confused. These associationsdon’t seem to have anything to do with the words you aresaying.Dr. Stew: That’s right.Hank: But then I don’t understand why people are sayingthem.Dr. Stew: Well, Hank, you have to remember. Everything’sassociative in a group.Hank: Oh, come on …Dr. Stew: Anyway, enough of that. Amanda, what’s beengoing on with you?

Amanda: Well, my father has been commuting back andforth between Boston and New York. He’s just home on theweekends.Dr. Stew: By himself? Does he drive?

Amanda: He usually rides with my uncle and theneighbor down the street. They have the same situation.Karen: That must be tough for you.Bill: You must miss him a lot.Hank: Isn’t someone going to say something about thecommutator subgroup, or the group being abelian becauseeveryone commutes?Dr. Stew: No. What does that have to do with anything?Hank: I am so confused.Dr. Stew: Let’s talk about that, Hank. It sounds like you arehaving an identity crisis.Hank: It does?Dr. Stew: Yes, it’s unclear who is the identity in your familygroup.Hank: I’m not following you.Dr. Stew: Who is it when multiplied by any other member ofthe family yields that same member of the family?Hank: You know, I am having some trouble interpretingthis analogy with a group. What exactly is multiplication oftwo people?Dr. Stew: Well, what would you like it to be?Hank: How about something I can understand, not justsome nebulous ill-defined concept created so this piss pooranalogy can be sustained ad nauseam.Dr. Stew: I sense some hostility from you.Hank: Well, yes, I am a little frustrated.Dr. Stew: Maybe we can figure out where this hostility iscoming from. It probably goes back a ways. Do you havesiblings?Hank: Yes. I have two older brothers, Jeff and Tom, and anolder sister Caroline. And then a much younger sister Liz.She was really brought up more by my three older siblingsthan by my parents.Dr. Stew: I see. Now, tell us. If you were going to give aword in the generators that are your mother, father, andsiblings that best describes you, what would it be?Hank: Excuse me?Dr. Stew: You know. Let M denote your mother, F yourfather, J for Jeff, T for Tom, C for Caroline and L for Liz. Thenmake a word from these generators and their inverses thatbest describes you, that encompasses what parts of youcome from these generators. We are all a product of ourfamilies.Hank: I would say you’re kidding, but I am guessing youare not. Okay, I’ll play along, How about … um …

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M F C-1JTC L-1C F -1M -1?Bill: Ha!Hank: What? What now?Bill: That’s the trivial word.Hank: No it’s not.Bill: Yes it is. You said before that Liz was a product of Jeff,Tom and Caroline so L equals JTC, meaning JTCL-1 is trivial.That was in the middle of your word. Once we trivializethat, the remaining generators and inverses cancel and theword collapses. So yes that is the trivial word.

Dr. Stew: Hank, I find it indicative of your feelings of selfloathing that you would pick a trivial word to describeyourself. Clearly an identity crisis, as you see yourself as theidentity element.Hank: (Stands up.) That’s it. I am out of here. You peopleare crazy.Dr. Stew: We don’t use the word ‘‘crazy,’’ Hank. We say‘‘topologically mixing.’’ But, you know, perhaps it is best ifyou leave. You aren’t ready for group therapy. I think youneed one-on-one help, probably on a continuous basis. Youshould see an analyst.

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