Answer to Question #281250 in Abstract Algebra for Himanshi

Question

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.

Answer 1

Solution:

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

Cyclic notations of given permutations are:

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

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

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

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

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

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

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

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

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