Question

The order of the element 5+(4) in the factor group z↓12/⟨4⟩
is
(​

Answer 1

SOLUTION

TO DETERMINE

$\sf{The \: order \: of \: the \:element \: \: 5 + < 4> in \: the \: factor \: group \: \: \mathbb{Z_{12}}/ < 4 > }$

EVALUATION

First we define the order of an element in a group

Order of an element in a group G is defined to be the least positive integer n such that

$\sf{ {a}^{n} = e}$

$\sf{Here \: the \: factor \: group \: \: is \: \: \mathbb{Z_{12} }/ < 4 > }$

We have to find the

$\sf{\: order \: of \: the \:element \: \: 5 + < 4> in \: the \: factor \: group \: \: \mathbb{Z_{12}}/ < 4 > }$

We have

$\sf{ < 4 > = \{ \: 0,4,8 \: \}}$

Now

$\sf{4( 5 + < 4> ) = \: < 4 > }$

$\sf{ \therefore \:The \: order \: of \: the \:element \: \: 5 + < 4> in \: the \: factor \: group \: \: \mathbb{Z_{12} }/ < 4 > \: is \: 4 }$

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