Find the order of each element of the group Z 7 - 0 - - - 7

G={ Z _ 7 {[ 0 ] #160;} #160;} ={[ 1 ] ,[ 2 ] ,[ 3 ] ,[ 4 ] ,[ 5 ] ,[ 6 ] #160;} The identity element is [ 1 ] O{[ 1 ] #160;} =1 ...

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Standard XII

Mathematics

Definition of Sets

Find the orde...

Question

Find the order of each element of the group $(Z_{7} - {[0]}, \cdot 7)$

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Solution

$G = {Z_{7} - {[0]}}$
$= {[1], [2], [3], [4], [5], [6]}$
The identity element is $[1]$
$O {[1]} = 1$
$O {[2]} = 3$ since ${[2]}^{3} = 1$
$O {[3]} = 6$ since ${[3]}^{6} = 1$
$O {[4]} = 3$ since ${[4]}^{3} = 1$
$O {[5]} = 6$ since ${[5]}^{6} = 1$
$O {[6]} = 2$ since ${[6]}^{2} = 1$

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Similar questions

If -1, $α$ , $α^{3}$ , $α^{5}$ , $¯ ¯¯ ¯ α$ , $^{3}$ , $^{5}$ are the roots of the equation $z^{7}$ + 1 = 0. Find the value of $c o s \frac{π}{7}$ $\times$ $c o s \frac{3 π}{7}$ $\times$ $c o s \frac{5 π}{7}$ .

Q. Let

z_{j}, j = 1, 2, 3, . ., 7

be the roots of the equation

z^{7} = (1 - z)^{7} .

Then the value of

7 \sum j = 1 R e (z_{j}),

where

R e (z)

denotes the real part of

z

, is

Q. Solutions of the equation

z^{7} - 1 = 0

are given by

Q. Determine the variables in the expression

x^{7} + 7 x y z + y^{7} + z^{7} + 7

Q. Find the number of terms in the expansion of

{(2 x + 3 y + z)}^{7}

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Find the order of each element of the group (Z7−{[0]},⋅7)

G={Z7−{[0]}}={[1],[2],[3],[4],[5],[6]}The identity element is [1]O{[1]}=1O{[2]}=3 since [2]3=1O{[3]}=6 since [3]6=1O{[4]}=3 since [4]3=1O{[5]}=6 since [5]6=1O{[6]}=2 since [6]2=1

Find the order of each element of the group $(Z_{7} - {[0]}, \cdot 7)$

$G = {Z_{7} - {[0]}}$
$= {[1], [2], [3], [4], [5], [6]}$
The identity element is $[1]$
$O {[1]} = 1$
$O {[2]} = 3$ since ${[2]}^{3} = 1$
$O {[3]} = 6$ since ${[3]}^{6} = 1$
$O {[4]} = 3$ since ${[4]}^{3} = 1$
$O {[5]} = 6$ since ${[5]}^{6} = 1$
$O {[6]} = 2$ since ${[6]}^{2} = 1$