# “Associativity, Left Identity, Left Inverse Elements” Makes a Group

By | Februar 9, 2009

In most undergraduate algebra textbooks, 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:
$$a(bc)=(ab)c$$ for all $$a,~b,~c \in G$$.
• Existence of an identity element:
There exists $$e\in G$$ s.t. $$ea = ae = a$$ for all $$a \in G$$.
• Existence of inverse elements:
For all $$a \in G$$ there exists $$a' \in G$$ s.t. $$a' a = a a' = e$$ where $$e$$ is the identity element.

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

Theorem. Let $$G$$ be a nonempty set with a binary operation and satisfy the following.

• Associative rule:
$$a(bc)=(ab)c$$ for all $$a,~b,~c \in G$$.
• Existence of a left identity element:
There exists $$e\in G$$ s.t. $$ea = a$$ for all $$a \in G$$.
• Existence of left inverse elements:
For all $$a \in G$$ there exists $$a' \in G$$ s.t. $$a' a = e$$ where $$e$$ is the left identity element.

Then $$G$$ is a group.

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 yields $$a a' = e$$ by the above lemma. 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.

Corollary. Let $$G$$ be a nonempty set with a binary operation and satisfy the following.

• Associative rule:
$$a(bc)=(ab)c$$ for all $$a,~b,~c \in G$$.
• Existence of a right identity element:
There exists $$e \in G$$ s.t. $$ae = a$$ for all $$a \in G$$.
• Existence of right inverse elements:
For all $$a \in G$$ there exists $$a' \in G$$ s.t. $$aa' = e$$ where $$e$$ is the right identity element.

Then $$G$$ is a group.