In Joseph Gallian's book Contemporary Abstract Algebra, he calls the above the "Socks-Shoes Property" for any group elements x and y.

**Exercise 1:**Put your socks on followed by your shoes. Now see if you can remove your socks without removing your shoes (or cutting your socks). If you can, then your shoes are too big!!

**Exercise 2:**Put your shoes on followed by your socks... now go to work/school like that. If no one says anything to you about it, then you must be hanging out with too many mathematicians!

Exercise 2 seems rather fun... but I'd probably get kicked out of school, made fun of, or have to wash my socks at the end of the day...

Not for me...Gallian would find it hilarious. I'm a grad student where Dr. Gallian teaches (University of Minnesota-Duluth).

No- for me it's ok, because it's early in the morning and I have to commute to get to work.

Also if you can do that, your socks are much too big... or very elastic.

Socks and shoes also provide a good illustration of the Axiom of Choice: if you have countably infinitely many pairs of shoes, you can choose one from each without AC, but if you have countably infinitely many pairs of socks then you can't choose one from each pair without AC. (You can specify the left shoe of each pair, but there is no such rule for the socks.)

It depends on the socks. Some have a definite left/right orientation.

Topologicaly, what's the difference between socks on and off?

XY \ne YX

You could remove your socks without removing your shoes by cutting the fronts of the shoes open.

@Jeremiah - then you'd have sandals. So that would be, uh, a homomorphism from shoes to their subgroup of sandals?

Hey Mike, can I invite you to do a comic strip guest post on my mathematics website?

Excuse my (probably stupid) question, but shouldn't X and Y be in a reverse order? Suppose F is a foot. If we want to put on socks we have to use XF and so on...

I always loved how profs would say that property. "The inverse of the product is the product of the inverse in reverse". It just rolls off the tongue.

I simply describe this as stacking clothing on feet. By the fact that is is a stack data structure, the last piece the mathematician put on must be the first one to be taken off. [/computerengineer]

I used Gallian during undergrad as well, totally remember this! :D

My teacher nevr explaind me abt dis socks shoes property while explaining inverse property.may b even he might not know it