Group Theory Discrete Mathematics


Qn) Z is a set of Integers, (Z,*) is a group with a*b= a+b+1, a,b ∈ Z. Then the inverse of a is
(i) -a
(ii) a+1
(iii) -2-a


First of all we need to find the identity element.
a*e=a, we need to find ‘e’. ----> equation(1)

We know a*b=a+b+1
Now a*e=a+e+1 ---->equation(2)

Now equating equation(1) & (2),

Now let ‘b’ be the inverse of ‘a’
We know a*b=e

So option (iii) is correct.


a*b = a + b + 1.

for inverse we need to find identity element.
identity is a.e = e.a = a

so e = a -1 .
since a - 1 + 1 = a.
so e = -1.

inverse is a.inverse = e.

so -a - 2

a - a - 2 + 1 = - 1.

so e is unique and inverse depends on element or e can be different for different a too in other groups ?? @Ruturaj


There is an Important property of the group. If Inverse exists then it is unique. There is a simple prove for this also.

a*(a^-1)=e -----> eqn(i)

Let ‘b’ be another inverse.Then a*b=e ---->eqn(ii)
Now let’s equate both the equations.


applying left cancellation rule,

we have a^(-1) = b

So this implies that the inverse is unique.


yaa unique in the sense for a particular element,but e is same irrespective of element or may be im missing something because in the question above ,inverse was -a - 2,so if a = 1,inverse id -3,a= 3 inverse is -5


Inverse of a=1 is -1-2= -3
Inverse of a = -3 is -(-3)-2 = 3-2 = 1


so inverse depends on which element im choosing and that will be unique,
How about identity element,does identity element also depends on which element we are doing the group operation


No, identity element is same for all elements in the group.
ab = e when a,b are inverse to each other.
d = e when c,d are inverse to each other .
here e is identity element.