Group Product/Examples/a x b = c

Examples of Operations on Product Elements

Solve for $x$ in:

$a x b = c$

Solution

\(\ds a x b\)

\(=\)

\(\ds c\)

\(\ds \leadsto \ \ \)

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

\(=\)

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

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

\(\ds \leadsto \ \ \)

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

\(=\)

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

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

\(\ds \leadsto \ \ \)

\(\ds x\)

\(=\)

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

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: Exercise $2 \ \text{(c)}$