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This document is an assignment from the Department of Mathematics at Jain College of Engineering, focusing on Group Theory as part of the Discrete Mathematical Structures course. It includes various questions related to group properties, such as proving abelian groups, defining groups, and demonstrating Lagrange's theorem. The assignment is intended for fourth-semester students and carries a maximum of 20 marks.

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Oliver Ryan Fernandes
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20 views2 pages

A (M5)

This document is an assignment from the Department of Mathematics at Jain College of Engineering, focusing on Group Theory as part of the Discrete Mathematical Structures course. It includes various questions related to group properties, such as proving abelian groups, defining groups, and demonstrating Lagrange's theorem. The assignment is intended for fourth-semester students and carries a maximum of 20 marks.

Uploaded by

Oliver Ryan Fernandes
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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Jain College of Engineering, Belagavi

Department of Mathematics
Assignment – 4
Module 5: Group Theory

Course Name: Discrete Mathematical Structures Course Code: BCS405A


Semester : IV Maximum Marks : 20
Date: 17/05/2025 Date of Submission:
Q.NO Question CO PO RBT
If * is a an operation on Z defined by xy=x+y+1, prove that (Z,*) is an abelian
1 5 1 L2
group.

2 Define Group.Show that fourth roots of unity is an abelian group under 5 1 L2

3 Define Klein 4 Group .Verify A= 1,3,5,7 is a Klein 4 group. 5 1 L2


Show that i) the identity of G is unique.
4 5 1 L2
ii) the inverse of each element of G is Unique
Show that (A, .) is an abelian group where A= a ∈ Q/a ≠− 1 and for any
5 5 1 L2
a,b ϵA, a. b =a+b+ab.
If G be a set of all non-zero real numbers and let a*b =ab/2 then show that(G, *) is
6 5 1 L2
an abelian group
7 Prove that intersection of two subgroups of a group G is also a subgroup of G. 5 1 L2
8 Prove that (z4 , +) ia cyclic group.Find all its generators. 5 1
If H,K are subgroups of a group G, prove that H∩ K Is also a subgroup of G. Is H ∪
9 5 1 L2
K A subgroup of G ?
If G be a group with subgroup H and K .If G = 660 and K = 66 and K∁H∁G and
10 5 1 L2
�ind the possible value for H
11 Prove that intersection of two subgroups of a group G is also a subgroup of G 5 1 L2
12 State and prove Lagrange’s theorem. 5 1 L2
1 2 3 4
Let G=S4 for α = Find the subgroup H= < α > determine the left
13 2 3 4 1 5 1 L2
cosets of H in G.
Define cyclic group and show that (G,*) whose multiplication table is as given
below is cyclic
* a b c d e f
a a b c d e f
14 b b c d e f a 5 1 L2
c c d e f a b
d d e f a b c
e e f a b c d
f f a b c d e
*CO- Course Outcome: Demonstrate the application of discrete structures in different fields of
computer science.
Prepared By Scrutinized By Approved By
Signature: Signature: Signature:
Name:Dr.Manjula Kalyanshetti Name: Prof. Umesh S. Mujumdar Name: Dr. Prashant V Patil

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