x*y=x-yLet the binary operation * be defined on Z by the given rule. Determine whether Z is a group with respect to and whether it is an abelian group. State which, if any, conditions fail to hold.

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You should check the four axioms of groups such that:

G1: closure: For `x, y in Z` , then `x * y in Z`

Notice that the difference of two integer numbers yields an integer number also, hence, the first axiom G1 holds.

G2: identity

You need to look for an element e such that:

`x*e = e*x = x`

You need to substitute `x - e`  for `x*e ` such that:

`x - e = x => -e = x - x => e = 0`

Hence, since there exists an identity element e = 0, then the second axiom G2 holds.

G3: inverses

You need to look for an element such that:

`x*x^-1 = x^(-1)*x = e`

`x - x^(-1) = 0 => x^(-1) = x`

Hence, since there exists an inverse element `x^(-1) = x` , then the third axiom G3 holds.

G4: associativity

You need to check if the identity `(x*y)*z = x*(y*z)`  holds such that:

`(x-y)*z = x*(y-z) => x - y - z = x - (y- z) => x - y - z = x - y + z`

Since the fourth axiom does not hold, hence, `(Z,*)`  does not represent a group.