Answer to Question #281250 in Abstract Algebra for Himanshi

Question #281250

Let A be {1, 2, 3, 4}. Then the sequences 124, 421, 341 and 243 are same  permutations of A taken 3 at a time. The sequences 12, 43, 31, 24, and 21 are examples of different permutations of A taken two at a time.


1
Expert's answer
2021-12-20T16:42:06-0500

Solution:

Given, A ={1, 2, 3, 4}

Cyclic notations of given permutations are:

(1 2 4)=(12342431)(1\ 2\ 4)=\begin{pmatrix} 1 &2&3&4 \\ 2& 4&3&1 \end{pmatrix}

(4 2 1)=(42132143)(4\ 2\ 1)=\begin{pmatrix} 4 &2&1&3 \\ 2& 1&4&3 \end{pmatrix}

(3 4 1)=(34124132)(3\ 4\ 1)=\begin{pmatrix} 3 &4&1&2 \\ 4& 1&3&2 \end{pmatrix}

(2 4 3)=(24314321)(2\ 4\ 3)=\begin{pmatrix} 2 &4&3&1 \\ 4& 3&2&1 \end{pmatrix}

(1 2)=(12342134)(1\ 2)=\begin{pmatrix} 1 &2&3&4 \\ 2& 1&3&4 \end{pmatrix}

(4 3)=(43123412)(4\ 3)=\begin{pmatrix} 4 &3&1&2 \\ 3& 4&1&2 \end{pmatrix}

(3 1)=(31241324)(3\ 1)=\begin{pmatrix} 3 &1&2&4 \\ 1& 3&2&4 \end{pmatrix}

(2 4)=(24314231)(2\ 4)=\begin{pmatrix} 2 &4&3&1 \\ 4& 2&3&1 \end{pmatrix}

(2 1)=(21341234)(2\ 1)=\begin{pmatrix} 2 &1&3&4 \\ 1& 2&3&4 \end{pmatrix}


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