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

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

.

That 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.

Thank you.

Hjörtur