GO Classes Test Series 2024 | Discrete Mathematics | Test 3 | Question: 16

GO Classes asked Apr 27, 2022 • retagged May 4, 2023 by Lakshman Bhaiya

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Consider the group $(\mathbf{Z}, +)$ where $\mathbf{Z}$ is set of all integers and “$+$” is addition operation.

Consider the following subset of $\mathbf{Z}:$

$\text{H} = {30x + 42y + 70z | x, y, z \in \mathbf{Z}}.$

Which of the following is true for $\text{H}?$

$\text{H}$ is a subgroup of $\mathbf{Z}.$
$\text{H}$ is Not a subgroup of $\mathbf{Z}$ because $\text{H}$ is not closed under $+.$
$\text{H}$ is Not a subgroup of $\mathbf{Z}$ because $\text{H}$ doesn’t have any identity element.
$\text{H}$ is Not a subgroup of $\mathbf{Z}$ because for some element $g$ in $\text{H},$ inverse of $g$ doesn’t belong to $\text{H}.$

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GO Classes asked Apr 27, 2022 • retagged May 4, 2023 by Lakshman Bhaiya

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Hence, $\text{H}$ is a subgroup.

GO Classes answered Apr 27, 2022 • edited Apr 27, 2022 by Lakshman Bhaiya

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sk91

commented Jan 17, 2023

@Pranavpurkar

Here, * is the binary operator on which the group is defined.

In a group, for all $x,y$ $\epsilon$ $G$, $x * y$ $\epsilon$ $G$ and this is the closure property. Same is expected to hold for a subgroup as well

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Pranavpurkar

commented Jan 17, 2023

@sk91

Thanks :)

yes this is the closure property only right? i got confused bcoz of “Product”.

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sk91

commented Jan 17, 2023

yes that’s correct.

As you mentioned, we have to be careful when dealing with * as we are accustomed to seeing it as product.

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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

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