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Algebra problem solved 3

Problem: "Show that if each element in a group is its own inverse then the group is abelian.

That's an easy one.

ab = e(ab)e = bb(ab)aa = b(ba)(ba)a = beea = ba

Algebra problem solved 2

Problem: "Prove that a finite monoid in which the cancellation law holds is a group. "

To prove it, we must show that for all a there is an inverse element.

We know that the monoid in question is finite so:

\forall a \exists n such that an = e where e is the identity element.

This stems from the fact that monoids are closed under their operation.

an= e implies an-1a= e. So the inverse element is an-1

Group of symmetries of a equilateral triangle

I was looking for the operation table for the symmetries of a equilateral triangle. I have found this webpage:
http://members.tripod.com/~dogschool/trianglegroup.html

  • e stand for the Do nothing movement.
  • a stand for the Rotate 120 degrees counterclockwise

    movement.

  • b stand for the Rotate 240 degrees counterclockwise
    movement.

Algebra problem solved

Problem: "Let A be an infinite set. Let us consider the set of functions \left{f \in A^A : f \ is\hspace{4}surjective\right} and the operation of function composition. Do they form a semigroup? A monoid? A group? Does the right cancellation law hold? What about the left cancellation law? And what happens if we consider the set of all function from A to A instead of the set of surjective functions?"

a) To prove that the set is a semigroup, we must show that associativity takes place, so:

(f \circ g) \circ h (x)= f \circ (g \circ h) (x)