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Random Talk on Random Thoughts

Weaker Group Axioms

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The following axioms seem to be weaker.

Let $G$ be a set, and $*: G \times G \to G$ be a binary operation.

$\forall a,b,c \in G, (a * b) * c = a * (b * c)$
Existence of left identity
$\exists e \in G \text{ s.t. } \forall a \in G, e * a = a$
Existence of left inverse
$\forall a \in G, \exists a^{-1} \in G \text{ s.t. } a^{-1} * a = e$

Claim: They’re actually equivalent definition of a group.


Existence of inverse
We try to show that a left inverse of a group element is also a right inverse of the group element.

$\forall a \in G,$

\[ \begin{split} a * a^{-1} &= e * (a * a^{-1}) &\qquad \text{(existence of left identity)}\\ &= ((a^{-1})^{-1} * a^{-1}) * (a * a^{-1}) &\qquad \text{(existence of left identity of $a^{-1}$)}\\ &= (((a^{-1})^{-1} * a^{-1}) * a) * a^{-1} &\qquad \text{(associativity)}\\ &= ((a^{-1})^{-1} * (a^{-1} * a)) * a^{-1} &\qquad \text{(associativity)}\\ &= ((a^{-1})^{-1} * e) * a^{-1} &\\ &= (a^{-1})^{-1} * (e * a^{-1}) &\qquad \text{(associativity)}\\ &= (a^{-1})^{-1} * a^{-1} &\\ &= e & \end{split} \]

$\forall a \in G, \exists a^{-1} \in G \text{ s.t. } a * a^{-1} = e = a^{-1} * a.$

Existence of identity
Similarly, we try to show that a left identity is also a right identity. $\forall a \in G,$
\[ \begin{split} a * e &= a * (a^{-1} * a) &\qquad \text{(existence of inverse of $a$)}\\ &= (a * a^{-1}) * a &\qquad \text{(associativity)}\\ &= e * a &\\ &= a& \end{split} \]

$\exists e \in G \text{ s.t. } \forall a \in G, a * e = e * a = a$.

An application in secondary school mathematics

In high school, to show that $B \in M_{n \times n}(\R)$ is an inverse of $A \in M_{n \times n}(\R)$1, one is taught to show both $AB = I$ and $BA = I$. Calculating $AB$ and $BA$ is hard in general.

Taking $G = \GL(n,\R)$ and regard $*$ as matrix multiplication, one can just show that $AB = I$ by direct calculation and then conclude that $BA = I$, and vice versa.

  1. To be more general, $\R$ can be replaced with $\F$.