Question

Suppose \( f, g, h \) are permutations of the set \( \{ \alpha, \beta, \gamma, \delta \} \), where
f interchanges \( \alpha \) and \( \beta \) but fixes \( \gamma \) and \( \delta \),
g interchanges \( \beta \) and \( \gamma \) but fixes \( \alpha \) and \( \delta \),
h interchanges \( \gamma \) and \( \delta \) but fixes \( \alpha \) and \( \beta \).
Which of the following permutations interchange(s) \( \alpha \) and \( \delta \) but fix(es) \( \beta \) and \( \gamma \)?

• A. \( f \circ g \circ h \circ g \circ f \)
• B. \( g \circ h \circ f \circ h \circ g \)
• C. \( g \circ f \circ h \circ f \circ g \)
• D. \( h \circ g \circ f \circ g \circ h \)

Hint:

To solve permutation problems, carefully analyze how each permutation affects the elements of the set and compose them step by step.

Answer 1

The Correct Option is A, D

\[f=(\alpha\ \beta),\qquad g=(\beta\ \gamma),\qquad h=(\gamma\ \delta)\]

We test each option by tracking the images of \(\alpha,\beta,\gamma,\delta\).

(A) \(f\circ g\circ h\circ g\circ f\)

\[\begin{aligned} \alpha &\xrightarrow{f} \beta \xrightarrow{g} \gamma \xrightarrow{h} \delta \xrightarrow{g} \delta \xrightarrow{f} \delta, \\ \delta &\xrightarrow{f} \delta \xrightarrow{g} \delta \xrightarrow{h} \gamma \xrightarrow{g} \beta \xrightarrow{f} \alpha, \\ \beta &\xrightarrow{f} \alpha \xrightarrow{g} \alpha \xrightarrow{h} \alpha \xrightarrow{g} \alpha \xrightarrow{f} \beta, \\ \gamma &\xrightarrow{f} \gamma \xrightarrow{g} \beta \xrightarrow{h} \beta \xrightarrow{g} \gamma \xrightarrow{f} \gamma. \end{aligned}\]

Thus (\(\alpha, \delta\)) with \(\beta,\gamma\) fixed.

(D) \(h\circ g\circ f\circ g\circ h\)

\[\begin{aligned} \\ \alpha &\xrightarrow{h} \alpha \xrightarrow{g} \alpha \xrightarrow{f} \beta \xrightarrow{g} \gamma \xrightarrow{h} \delta, \\ \delta &\xrightarrow{h} \gamma \xrightarrow{g} \beta \xrightarrow{f} \alpha \xrightarrow{g} \alpha \xrightarrow{h} \alpha, \\ \beta &\xrightarrow{h} \beta \xrightarrow{g} \gamma \xrightarrow{f} \gamma \xrightarrow{g} \beta \xrightarrow{h} \beta, \\ \gamma &\xrightarrow{h} \delta \xrightarrow{g} \delta \xrightarrow{f} \delta \xrightarrow{g} \delta \xrightarrow{h} \gamma. \end{aligned}\]

Thus (\(\alpha, \delta\)) with \(\beta,\gamma\) fixed.

\[\boxed{\text{Correct options are (A) and (D).}}\]