I am studying by my self Basic Algebra by Nathan Jacobson, and I am currently doing the exercises from chapter 1.
I would be very thankful if someone would look over my answers and
tell me if my argument is sound.
1. Let be a monoid with multiplication p, and let . Define a new product in by .
Show that this defines a semigroup. Under what conditions on do we have a unit relative to ?
For this to be a semigroup then we have to show that is associative. Let denote .
Let , then
, because composition under is associative, so is associative so this is a semigroup.
Now we do have a unit relative to if and only if has an inverse under . Then the unit relative to is .
and for all
2. Show that cannot be the union of two proper subgroups.
Lets assume that is the union of proper subgroups . Since they are proper there exist elements , such that and .
Now since is a group then it is closed under composition, that is . Then or .
Since is a subgroup and then and is closed under composition. Then but that is a contradiction. In the same way
leads to which is also a contradiction.
Therefore cannot be the union of two proper subgroups.