Hw4 Problem 9

Problem 9

(*) The following problem shows that we can loosen the requirements in the last two group axioms and still get a group. Consider a set $G$ with a binary operation $*$ such that

• $*$ is associative
• there exists a left identity element $e \in G$ such that $e * x = x$ for all $x \in G$
• for each $a \in G$ there exists a left inverse $a' \in G$ such that $a' * a = e$.

(a) Show that left cancellation holds. That is, if $a * b = a * c$, then $b = c$.
(b) Show that the left identity element $e$ is also a right identity element for all $x \in G$.
(c) Show that the left inverse $a'$ for $a$ is also a right inverse for $a$.

Solution

For all $a, a', b, c, x, x', e \in G$.

(a)

(1)
$$a * b = a * c$$
(2)
$$a' * (a * b) = a' * (a * c)$$
(3)
$$(a' * a) * b = (a' * a) * c$$
(4)
$$e * b = e * c$$
(5)
$$b = c$$

So left cancellation holds.

(b)

(6)
$$x' * (x * e) = (x' * x) * e$$
(7)
$$= e * e$$
(8)
$$= e$$
(9)
$$= x' * x$$

Since

(10)
$$x' * (x * e) = x' * x$$
(11)
$$x * e = x$$

Then $e$ is a right identity element for all $x \in G$.

(c)

(12)
$$a' * (a * a') = (a' * a) * a'$$
(13)
$$= e * a'$$
(14)
$$= a'$$
(15)
$$= a' * e$$

Since

(16)
$$a' * (a * a') = a' * e$$
(17)
$$a * a' = e$$

Then $a'$ is a right inverse for $a$.