Division Laws for Groups
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Theorem
Let $G$ be a group.
Let $a, b, x \in G$.
Then:
- $(1): \quad a x = b \iff x = a^{-1} b$
- $(2): \quad x a = b \iff x = b a^{-1}$
Proof
All derivations can be achieved using applications of the group axioms:
Proof of $(1)$
| \(\ds a x\) | \(=\) | \(\ds b\) | by hypothesis | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds a^{-1} \paren{a x }\) | \(=\) | \(\ds a^{-1} b\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \paren{a^{-1} a } x\) | \(=\) | \(\ds a^{-1} b\) | Group Axiom $\text G 1$: Associativity | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds e x\) | \(=\) | \(\ds a^{-1} b\) | Definition of Inverse Element | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds x\) | \(=\) | \(\ds a^{-1} b\) | Definition of Identity Element |
and the converse:
| \(\ds x\) | \(=\) | \(\ds a^{-1} b\) | by hypothesis | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds a x\) | \(=\) | \(\ds a \paren{a^{-1} b }\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds a x\) | \(=\) | \(\ds \paren{a a^{-1} } b\) | Group Axiom $\text G 1$: Associativity | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds a x\) | \(=\) | \(\ds e b\) | Definition of Inverse Element | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds a x\) | \(=\) | \(\ds b\) | Definition of Identity Element |
$\blacksquare$
Proof of $(2)$
| \(\ds x a\) | \(=\) | \(\ds b\) | by hypothesis | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \paren{x a } a^{-1}\) | \(=\) | \(\ds b a^{-1}\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds x \paren{a a^{-1} }\) | \(=\) | \(\ds b a^{-1}\) | Group Axiom $\text G 1$: Associativity | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds x e\) | \(=\) | \(\ds b a^{-1} b\) | Definition of Inverse Element | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds x\) | \(=\) | \(\ds b a^{-1}\) | Definition of Identity Element |
and the converse:
| \(\ds x\) | \(=\) | \(\ds b a^{-1}\) | by hypothesis | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds x a\) | \(=\) | \(\ds \paren{b a^{-1} } a\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds x a\) | \(=\) | \(\ds b \paren{a^{-1} a }\) | Group Axiom $\text G 1$: Associativity | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds x a\) | \(=\) | \(\ds b e\) | Definition of Inverse Element | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds x a\) | \(=\) | \(\ds b\) | Definition of Identity Element |
$\blacksquare$
Sources
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 35.2$: Elementary consequences of the group axioms