MTH633 ASSIGNMENT NO. 1 FALL 2022
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MTH633 ASSIGNMENT NO. 1 FALL 2022 || 100% RIGHT SOLUTION || GROUP THEORY|| BY VuTech
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Assignment No.
1 MTH633 (Fall 2022)
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Q1:
Show that set of integers Z is not a group under the binary operation
multiplication.
Justify each of the group properties that holds or not. Marks = 5
Solution:
Here,
The set of integers Z is not a group
under the binary operation multiplication.
Now, we have
to justify each of the group properties that holds or not.
Lets suppose
a = 1, b = 2 and c = 3 where `a, b \in Z`
So,
- Commutative
Property
First, we discuss commutative property `\forall `a,b \in Z`
a * b = 1 * 2 = 2 = 2 * 1 = b * a
So, the commutative property is satisfied.
- Associative Property
Now, we discuss associative property `\forall a,b,c \in Z`
`a * (b*c) = 1 * (2*3) = 6 = (1*2) * 3 = (a*b)*c`
So, the associative property is satisfied.
- Identity Property
Now, we discuss identity property `\forall a \in Z`
`e*a = 1 * 1= 1 = 1 * 1 = a * e where \e = 1 \in Z`
So, the identity property is satisfied.
- Inverse Property
Now, we discuss inverse property `\forall b \in Z`
`b = \frac{1}{b} \to 2 \ne\frac{1}{2}`
So, the inverse property is not satisfied.
So, its prove
The set of
integers Z is not a group under the binary operation multiplication.
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