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

Examples of Operations on Product Elements

Solve for $x$ in:

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

Solution

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

\(=\)

\(\ds e\)

\(\ds \leadsto \ \ \)

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

\(=\)

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

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

\(\ds \leadsto \ \ \)

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

\(=\)

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

Product of both sides with $a$

\(\ds \leadsto \ \ \)

\(\ds x\)

\(=\)

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

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

\(\ds \leadsto \ \ \)

\(\ds x\)

\(=\)

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

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

\(\ds \leadsto \ \ \)

\(\ds x\)

\(=\)

\(\ds e\)

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 $2 \ \text{(a)}$