Question

Which of the following sets G with the indicated binary operation are groups?
1) G is the set of all integers and the operation is given by a∗b=2a+b.
2) G is the set of all integers and the operation is given by a∗b=a+b−1.
3) G is the set of complex numbers of absolute value 2 under multiplication.
4) G is set of all 3×3 matrices with integer entries whose determinant is 1, under multiplication.​

Answer 1

Answer:

option 2 , correct

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Answer 2

Option 2) G is the set of all integers and the operation is given by a∗b=a+b−1.

is the correct answer.

because we know that,

A binary operation:  ∗ on Zn by a ∗ b = a + b if a + b<n, and a ∗ b = a + b−n otherwise. Then Zn is an abelian group with respect to ∗, with identity 0. The inverse of a > 0 in Zn is n − a (the inverse of 0 is 0). This group is called the cyclic group of order n.