Determine whether or not the following system is a groupSolu

Determine whether or not the following system is a group:

Solution

For this set with this operation to form a group we need to have an identity element

Let, k be the identity element

So, a*k=-ak=a ,where a is any non zero integer

This gives k=-1

So,k=-1 is the identity element

So the set has an identity element

Now we need to find an inverse for each element

So for any a to have an inverse ,b we need

ab=-1

But,ab=-1 with a and b being non zero integers is only possibel if one of a and b is equal to 1 and other is equal to -1.

Hence not all elements have an inverse.

HEnce not a group.