“Associativity, Left Identity, Left Inverse Element” Makes a Group

In the most undergraduate algebra books, you can find the definition of a group as following:

A nonempty set \(G\) with a binary operation satisfying the following conditions is called a group:

• Associative rule. i.e. \(a(bc)=(ab)c\) for all \(a,~b,~c \in G\).
• Existence of identity. i.e. There exists \(e\in G\) s.t. \(ea = ae = a\) for all \(a \in G\).
• Existence of inverse element. i.e. For all \(a \in G\) there exists \(a' \in G\) s.t. \(a' a = a a' = e\).

But these three conditions for a set to be a group can be weaken.

Proof. Denote \(a'\) for the left inverse element of \(a \in G\).

First, as a lemma, we see that \(c^2 = c\) implies \(c = e\), for \[c = ec = (c' c) c = c' c^2 = c' c = e. \]Now let \(a\) be an arbitrary element in \(G\). We have to show that \(a'\), the left inverse element of \(a\), is the right inverse element and that \(e\), the left identity element, is the right identity element. Observe that\[(a a')^2 = (a a' )(a a' ) = a ((a' a) a' ) = a (e a' ) = a a' \]which implies \(a a' = e \) by the lemma above. Thus we conclude that \(a' \) is the right inverse element of \(a\). Besides, we have\[ae = a(a' a) = (a a' ) a = ea .\]Thus we conclude that \(e\) is the right identity element.