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