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

By | February 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.

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