Group Product/Examples/b x a^-1 = a^-1 b

Examples of Operations on Product Elements

Solve for $x$ in:

$b x a^{-1} = a^{-1} b$

Solution

\(\ds b x a^{-1}\)

\(=\)

\(\ds a^{-1} b\)

\(\ds \leadsto \ \ \)

\(\ds b x a^{-1} a\)

\(=\)

\(\ds a^{-1} b a\)

Product of both sides with $a$

\(\ds \leadsto \ \ \)

\(\ds b^{-1} b x a^{-1} a\)

\(=\)

\(\ds b^{-1} a^{-1} b a\)

Product of both sides with $b^{-1}$

\(\ds \leadsto \ \ \)

\(\ds x\)

\(=\)

\(\ds b^{-1} a^{-1} b a\)

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

$\blacksquare$

Sources

• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $3$: Elementary consequences of the definitions: Proposition $3.3$: Remark 1