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Groups with elements other than numbers,Permutation and Matrix group,Example

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  • Things to remember

Groups with elements other than numbers

Since a set may consist of any kind of non-material or material objects and a binary operation can also be defined in various ways, we can define groups whose elements are non-numerical or blocks of numbers or matrices, or physical movements like translation, rotation etc. we now discuss simple cases that involve blocks of numbers or non-numerical objects or processes.

  • Permutation group

Suppose we toss a coin. Then we either get head H or a tail T. Consider the set G = {H, T} and the ordered rearrangements or permutations of its elements. Obviously, only two cases arise:

$$H\;T\;or\;T\;H\;$$

In other words, we may replace H by H and tail T by T in the first case and replace head H by tail T and tail T by head H in the second case. This situation is represented symbolically in the following way :

$$\;\sigma\;=\;\begin{pmatrix}H&T\\H&T\end{pmatrix}\;\;or\;\sigma\;=\begin{pmatrix}1&2\\H&T\end{pmatrix}\;\;$$

$$\;and\;\mu\;=\;\begin{pmatrix}H&T\\T&H\end{pmatrix}\;\;or\;\mu\;=\;\begin{pmatrix}1&2\\T&H\end{pmatrix}\;\;$$

$$Consider\;the\;set\;S_G\;=\;\{\;\sigma\;,\;\mu\;\}\;and\;define\;the\;binary\;operation\;*\;on\;S_G\;as\;the\;product\;or\;composition\;’\sigma\mu\;’\;or\;’\sigma\;*\;$$ $$\mu\;of\;the\;two\;permutations\;as\;’\;One\;permutation\;\sigma\;followed\;by\;another\;permutation\;\mu\;or\;’\;Do\;\sigma\;then\;\mu\;.$$

$$\;We\;can\;easily\;show\;that\;*\;is\;associative\;\;that\;has\;\sigma\;=\begin{pmatrix}H&T\\H&T\end{pmatrix}\;$$

$$\;as\;it\;identity\;element\;,\;and\;each\;element\;is\;the\;inverse\;of\;itself.\;In\;particular\;$$

$$\;\sigma\;*\;\sigma\;=\;\begin{pmatrix}H&T\\H&T\end{pmatrix}\;\;*\;\begin{pmatrix}H&T\\H&T\end{pmatrix}\;\;=\;\begin{pmatrix}H&T\\H&T\end{pmatrix}\;\;=\;\sigma\;$$

$$which\;shows\;\sigma\;is\;both\;an\;identity\;element\;amd\;inverse\;of\;itself\;.$$

$$\;\sigma\;*\;\mu\;=\;\begin{pmatrix}H&T\\H&T\end{pmatrix}\;\;*\;\begin{pmatrix}H&T\\T&H\end{pmatrix}\;\;=\;\begin{pmatrix}H&T\\T&H\end{pmatrix}\;\;=\;\mu\;$$

$$which\;confirms\;\sigma\;is\;an\;identity\;element\;$$

$$\;\mu\;*\;\mu\;=\;\begin{pmatrix}H&T\\T&H\end{pmatrix}\;\;*\;\begin{pmatrix}H&T\\T&H\end{pmatrix}\;\;=\;\begin{pmatrix}H&T\\H&T\end{pmatrix}\;\;=\;\mu\;$$

$$\;which\;shows\;that\;\mu\;is\;the\;inverse\;of\;itself\;$$

Also, we can get that

$$(\sigma\;*\sigma\;)\;*\;\mu\;=\;\sigma\;*\;(\sigma\;*\;\mu\;)$$

All these show that S­G­ is a group under composition or production operation *. This group is a permutation group.

$$A\;binary\;operation\;\circ\;called\;product\;or\;multiplication\;of\;permutation\;is\;defined\;by\;the\;following\;way:$$

$$\alpha\;\circ\;\beta\;=\begin{pmatrix}a&b&c\\b&c&a\end{pmatrix}\;\circ\;\begin{pmatrix}a&b&c\\c&a&b\end{pmatrix}\;=\;\begin{pmatrix}a&b&c\\a&b&c\end{pmatrix}\;$$

Matrix group:

The set of square matrices of a given order is known to possess the associative property, the zero matrices as the identity matrix and the negative of a matrix as the inverse of the matrix. This means the set of all the squares matrix group of the given order. A matrix group of order 2 under matrix addition is generally denoted by:

$$M_2\;=\;(\begin{pmatrix}a&b\\c&d\end{pmatrix}\;,+)\;\;\;\;\;\;\;where\;a,\;b,\;c,\;d\;\in\;R$$

Furthermore, the set of non-singular square matrices of a given order under matrix multiplication is known to be associative, has the unit matrix as the identity matrix, and the inverse of the given matrix as the inverse element. This means the set of non-singular square matrices forms a group under the operation matrix multiplication.

Example (Permutation Group):

$$Given\;the\;permutation\;group\;(S,\;*)\;with\;S\;=\;\{\varepsilon,\;\sigma_1\;\sigma_2\;\sigma_3,\;\sigma_4\;\sigma_5\;\}\;$$

$$where$$

$$\varepsilon\;=\;\begin{pmatrix}1&2&3\\1&2&3\end{pmatrix},\;\sigma_1\;=\;\begin{pmatrix}1&2&3\\2&3&1\end{pmatrix}\;\sigma_2\;=\;\begin{pmatrix}1&2&3\\3&1&2\end{pmatrix}$$

$$\;\sigma_3\;=\;\begin{pmatrix}1&2&3\\1&3&2\end{pmatrix}\;,\;\;\sigma_4\;=\;\begin{pmatrix}1&2&3\\3&2&1\end{pmatrix}\;and\;\;\sigma_5\;=\;\begin{pmatrix}1&2&3\\2&1&3\end{pmatrix}$$

Show that

$$\varepsilon\;is\;he\;identity\;element\;under\;product\;of\;permutation,$$

$$\sigma_2\;\circ\;\sigma_3\;\in\;S,$$

$$\sigma_1\;and\;\sigma_2\;are\;the\;inverses\;of\;each\;other\;and\;they\;satisfy\;commutative\;property,\;$$

$$\sigma_1\;\sigma_2\;and\;\sigma_3\;satisfy\;associativity.\;$$

Solution:

$$\varepsilon\;*\;\sigma_1\;=\;\begin{pmatrix}1&2&3\\1&2&3\end{pmatrix}\;*\;\begin{pmatrix}1&2&3\\2&3&1\end{pmatrix}=\begin{pmatrix}1&2&3\\2&3&1\end{pmatrix}\;=\;\sigma_1$$

$$\varepsilon\;*\;\sigma_2\;=\;\begin{pmatrix}1&2&3\\1&2&3\end{pmatrix}\;*\;\begin{pmatrix}1&2&3\\3&1&2\end{pmatrix}=\begin{pmatrix}1&2&3\\3&1&2\end{pmatrix}\;=\;\sigma_2$$

$$\therefore\;\varepsilon\;is\;the\;identity\;element.$$

Now,

$$\sigma_2\;\circ\;\sigma_3\;=\;\begin{pmatrix}1&2&3\\3&1&2\end{pmatrix}\;\circ\;\begin{pmatrix}1&2&3\\1&3&2\end{pmatrix}=\begin{pmatrix}1&2&3\\2&1&3\end{pmatrix}\;=\;\sigma_5\;\in\;S$$

Again,

$$\sigma_1\;*\;\sigma_2\;=\;\begin{pmatrix}1&2&3\\2&3&1\end{pmatrix}\;\circ\;\begin{pmatrix}1&2&3\\3&1&2\end{pmatrix}=\begin{pmatrix}1&2&3\\1&2&3\end{pmatrix}\;=\;\varepsilon$$

$$\sigma_2\;*\;\sigma_1\;=\;\begin{pmatrix}1&2&3\\3&1&2\end{pmatrix}\;\circ\;\begin{pmatrix}1&2&3\\2&3&1\end{pmatrix}=\begin{pmatrix}1&2&3\\1&2&3\end{pmatrix}\;=\;\varepsilon$$

$$\therefore\;\sigma_1\;and\;\sigma_2\;are\;the\;inverses\;of\;each\;other.\;$$

$$Also,\;\sigma_1\;*\;\sigma_2\;=\;\sigma_2\;*\;\sigma_1\;$$

$$\therefore\;\sigma_1\;and\;\sigma_2\;follow\;commutative\;property\;$$

Taken reference from

( Basic mathematics Grade XII and A foundation of Mathematics Volume II and Wikipedia.com )



  • The definition and concept of the permutation and matrix group are very important.
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