The Gateway to Computer Science Excellence
+1 vote

Let $G =\{a_1,a_2, \dots ,a_{12}\}$ be an Abelian group of order $12$ . Then the order of the element $ ( \prod_{i=1}^{12} a_i)$ is

  1. $1$
  2. $2$
  3. $6$
  4. $12$
in Set Theory & Algebra by Loyal (7.4k points)
edited by | 242 views

1 Answer

+2 votes
Best answer
Every element is multiplied with its inverse. The identity element is its own inverse. That leaves us with $11$ elements other than identity. So, one element (except the identity element) is its own inverse. The other $10$ elements get cancelled out as they are multiplied with their respective inverse elements. So, the order of the element $\prod_{i=1}^{12}a_i$ is $2$.

Option $(B)$ is correct.
by Active (3.4k points)
selected by
will it not be 6??

It is told summation of all elements
$\prod_{i=1}^{12}a_i$ is product symbol. How summation will be 6? Didn't understand. :)
yes, product

$i=1$, So elements are  $a_{1}$

$i=2$, So elements are $\left \{ a_{1},a_{2} \right \}$

$i=3$, So elements are $\left \{ a_{1},a_{2},a_{3} \right \}$


Now, $G=\left \{ a_{1},a_{2},a_{3},............,a_{12} \right \}$, here order of group is $12$, So order of it's subgroup


Now if  we do product of these elements , it will be $a_{1}.$$\left \{ a_{1},a_{2} \right \}$$.\left \{ a_{1},a_{2},a_{3} \right \}........$$\left \{ a_{1},a_{2},a_{3},............,a_{12} \right \}$

So, order will be $1,2,3,6,12$

Shouldn't the product be $a_1.a_2.a_3...a_{12}$? I don't understand how you are calculating the product. Explain a bit more.
Why did you write this: $$a_1.\{a_1, a_2\}.\{a_1, a_2, a_3\}.....\{a_1, a_2, a_3, ..., a_{12}\}$$
U mean $G$ same as $\prod a_{i} ??$

answer will be $12$


@Shikha Mallick

what about this?

The order of an element is defined as "m" such that $a^{m}$ is equal to the identity.

So, the given product evaluates to some $(a')^{2}$ where $a' $ is the element which is its own inverse.
So that means that $(a')^{2}$ is equal to identity. Hence, order of $(a')$ is 2. Therefore, the order of the product is 2.

Related questions

Quick search syntax
tags tag:apple
author user:martin
title title:apple
content content:apple
exclude -tag:apple
force match +apple
views views:100
score score:10
answers answers:2
is accepted isaccepted:true
is closed isclosed:true
50,737 questions
57,313 answers
105,046 users