Keep Groups Out of the Bedroom

This happened when I was a mathematician in a former life.

‘I want to turn the mattress,’ my wife said after dinner.

‘Okay.’

‘Furthermore, I want the head and foot swapped, the left and right sides swapped, and the top and bottom swapped.’

‘Okay. Let me finish my drink.’

She came back in the middle of Wild Girls on tv and said, ‘Have you finished your drink?’

‘Yes. Look at this.’ I handed her a piece of paper on which I had scribbled

                            

  

 ‘What is this?’

‘This tells you that you cannot do what you said.’

‘My ass,’ she said, ‘it doesn’t tell me anything.’

‘Au contraire, ma chère.  Take this paper and go study the mattress. The letter x denotes the act of flipping the mattress head-to-foot, y the flipping of it side-to-side, and z rotating it in a horizontal plane one hundred and eighty degrees. In this multiplication table, for example, xy means first doing y and then doing x, or, which turns out to be the same, doing x and then y. The table says, for example, that doing x and then y (or y and then x) leaves the mattress in the same state - relative to head/foot, left/right sides, and top/bottom - as just doing z.

‘Furthermore, doing x twice, or y, or z, leaves the mattress in the original state. The capital eye stands for identity, that is, no resulting change.’

‘Are you drunk?’

‘Pas du tout, mon petit chou. Go. Study.’

She stomped out.

Returning a while later, she asked, ‘Okay, so how does this - thing - prove I can’t do what I said?’

‘Consider. Clearly x leaves the left side on the left and the right side on the right, right? And y leaves the head at the head and the foot at the foot? And z leaves the top on top and the bottom on bottom?’

Nodding, nodding, nodding.

‘So that’s it. It’s obvious from the multiplication table, which can easily be verified mentally, that x, y, and z are all there is. It’s a closed system, and what you want ain’t there.’

‘Actually,’ I rattled on, ‘this is a tiny example of what we mathematicians call a group. The fact that the order of doing x and then y, or y and then x, produces the same result says this is a commutative group.  It’s actually a subgroup of a larger non-commutative group called the . . .’

‘Stop!’ she cried. ‘Keep your groups out of our bedroom.’ Pouring herself a drink she added, ‘And here’s one damn thing your little group doesn’t tell you: I want a new mattress.’