GATE CSE
First time here? Checkout the FAQ!
x
+4 votes
752 views

Consider the set S = $\{1, ω, ω^2\}$, where $ω$ and $ω^2$ are cube roots of unity. If * denotes the multiplication operation, the structure (S, *) forms

  1. A Group
  2. A Ring
  3. An integral domain
  4. A field
asked in Set Theory & Algebra by Veteran (13.5k points) 155 263 327
edited by | 752 views

3 Answers

+11 votes
Best answer

Answer: A

Cayley Table
  1 ω ω2
1 1 ω ω2
ω ω ω2 1
ω2 ω2 1 ω

The structure (S,*) satisfies closure property, associativity, commutativity. The structure also has an identity element (i.e. 1) and an inverse for each element. So, the structure is an abelian group.

answered by Veteran (35.4k points) 52 204 346
Can anyone explain what are ring, field, and integral domain and why the given structure is not one of them?

Rajarshi Sarkar can you please explain why it is not a ring ?

+3 votes
Its satisfies 1.closure property 2.associativity 3.identity element exists i.e 1 4.inverse element exist for each element So it is a group to attack these type of questions draw the composition table then it wl be easier ...
answered by Veteran (13.6k points) 15 70 142
+1 vote

In this question, only one binary operation is given so it cannot be a Ring or Integral domain or field.For Ring or Integral domain or for field there must be Two binary operations are required and the algebraic structure looks like (S ,+ ,⨉) .

So ,option b ,c and d are False.

 Now check the properties of groups.It satisfies all the conditions of groups.

Hence, it is a Group.

The correct answer is,(A) A Group

answered by Boss (8.8k points) 3 8 12


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
Top Users Oct 2017
  1. Arjun

    23210 Points

  2. Bikram

    17018 Points

  3. Habibkhan

    6652 Points

  4. srestha

    5864 Points

  5. Debashish Deka

    5430 Points

  6. jothee

    4908 Points

  7. Sachin Mittal 1

    4762 Points

  8. joshi_nitish

    4274 Points

  9. sushmita

    3954 Points

  10. Silpa

    3698 Points


Recent Badges

Regular Juhi Sehgal
Popular Question vineet.ildm
Nice Comment Arjun
100 Club vipul verma
Notable Question jothee
Popular Question jothee
Nice Question shivangi5
Regular rinks5
Notable Question shipra tressa
Regular sasi
27,247 questions
35,056 answers
83,703 comments
33,183 users