Question

Which of the following is not a group?

• A. $(Z,\bullet )$
• B. $(Z_n,{+}_n)$
• C. $(R,+)$
• D. $(Z,+)$

Answer 1

Answer: (A)

$(Z,\bullet )$

The non-zero integers under multiplication has an identity element $1$.However, not every integer has a multiplicative inverse.So this does not have multiplicative inverse.Hence it is not a group.

The set of integers is a group under the OPERATION of addition:The integers under the OPERATION of addition are CLOSED, ASSOCIATIVE, have IDENTITY $0$, and that any integer $x$ has the INVERSE $-x$. Because the set of integers under addition satisfies all four group PROPERTIES, it is a group

The set of real numbers is a group with respect to addition as the group is isomorphic to the set of real numbers without $0$ with respect to multiplication. What's more is that this group is abelian, that means any two elements will commute with respect to addition.
A group is a set equipped with a binary operation that combines any two elements to form a third element in such a way that four conditions called group axioms are satisfied, namely closure, associativity, identity and invertibility. Only option A does not satisfy this definition. Hence, option $\left(A\right)$ is not a Group.