Group Product/Examples/a x a^-1 = a
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Examples of Operations on Product Elements
Solve for $x$ in:
- $a x a^{-1} = a$
Solution
\(\ds a x a^{-1}\) | \(=\) | \(\ds a\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds a^{-1} a x a^{-1}\) | \(=\) | \(\ds a^{-1} a\) | Product of both sides with $a^{-1}$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds a^{-1} a x a^{-1} a\) | \(=\) | \(\ds a^{-1} a a\) | Product of both sides with $a$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds x\) | \(=\) | \(\ds a^{-1} a a\) | Group Axiom $\text G 3$: Existence of Inverse Element | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds x\) | \(=\) | \(\ds a\) | 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: Exercise $2 \ \text{(b)}$