Inverse of Group Product/Proof 1

Theorem

Let $\struct {G, \circ}$ be a group whose identity is $e$.

Let $a, b \in G$, with inverses $a^{-1}, b^{-1}$.

Then:

$\paren {a \circ b}^{-1} = b^{-1} \circ a^{-1}$

$\ds \paren {a \circ b} \circ \paren {b^{-1} \circ a^{-1} }$

$=$

$\ds \paren {\paren {a \circ b} \circ b^{-1} } \circ a^{-1}$

Group Axiom $\text G 1$: Associativity

$\ds $

$=$

$\ds \paren {a \circ \paren {b \circ b^{-1} } } \circ a^{-1}$

Group Axiom $\text G 1$: Associativity

$\ds $

$=$

$\ds \paren {a \circ e} \circ a^{-1}$

Group Axiom $\text G 3$: Existence of Inverse Element

$\ds $

$=$

$\ds a \circ a^{-1}$

Group Axiom $\text G 2$: Existence of Identity Element

$\ds $

$=$

$\ds e$

Group Axiom $\text G 3$: Existence of Inverse Element

$\paren {a \circ b} \circ \paren {b^{-1} \circ a^{-1} } = e \implies \paren {a \circ b}^{-1} = b^{-1} \circ a^{-1}$

$\blacksquare$

1964: Walter Ledermann: Introduction to the Theory of Finite Groups (5th ed.) ... (previous) ... (next): Chapter $\text {I}$: The Group Concept: $\S 2$: The Axioms of Group Theory
1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 4.6$. Elementary theorems on groups: Theorem $\text{(iv)}$
1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 1.4$: Theorem $2$
1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: The Group Property
1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: The Definition of Group Structure: $\S 28 \ (3)$
1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 35.6$: Elementary consequences of the group axioms
1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $3$: Elementary consequences of the definitions: Corollary $3.4$