Question

Determine whether or not ∗ gives a group structure on the set. If it is not a group, say which

axioms fail to hold.

Define ∗ on Z by a ∗ b = ab.

Answer 1

Consider the operation $\ast$ on $Z$ defined by

$a\ast b=ab$

Require to determine whether or not $\ast$ gives a group structure on the set $Z$

Let us verify the group axioms:

(1) Closure: $a,b\in Z\Rightarrow a*b\in Z$

Now $a,b\in Z\Rightarrow ab\in Z$

$\Rightarrow a* b\in Z$

So, $a,b\in Z\Rightarrow a*b\in Z$

Therefore, $Z$ satisfies the closure axiom under the operation $\ast$

(2) Associativity: $a,b,c\in Z\Rightarrow a* (b* c)=(a* b)\star c$

Now $a,b,c\in Z\Rightarrow a* (b* c)=a* (bc)=a(bc)=abc$

And

$(a* b)* c=(ab)* c=(ab)c=abc$

So, $a,b,c\in Z\Rightarrow a* (b* c)=(a*b)* c$

Therefore, $Z$ satisfies the associativity axiom under the operation $\ast$

(3) Identity: For each $a\in Z$ there exists $e\in Z$ such that $a*e=a=e*a$

Now $a*e=a\Rightarrow ae=a$

$\Rightarrow e=1\in Z$

And $a*1=a(1)=a=1(a)=1*a$

Therefore, $1$ is the identity in $Z$

(4) Inverse: For each non zero $a\in Z$ there exists $b\in Z$ such that $a*b=e=b*a$

Let $a$ be a non zero integer

Now $a*b=1\Rightarrow ab=1$

$\Rightarrow b=\frac{1}{a}\notin Z$

Therefore, inverse axiom fail to hold

Hence, $(Z,*)$ is not a group.