Q1 State the first of the 3 group axioms G1G2G3 that fails t

Q1. State the first of the 3 group axioms (G1,G2,G3) that fails to hold and justify your claim.

a. N = {1,2,3....}

b = R^x (nonzero reals) under * given by x*y =x/y.

c. R under * given by x*y = |xy|.

Solution

a) Given that N = { 1,2,3,... }

Then N is not a group with respect to addition since there is no additive inverse 0 in N.

and N is not a group with respect to multiplication since there is no multiplicative inverse for the element 2 in N.

b) Let R^x (nonzero reals) under the operation * on R^x defined by x*y =x/y.

Then R^xis not a group Since * is not associative as

x*(y*z) = x*(y/z) = x/(y/z) = xz/y and (x*y)*z =(x/y)*z = (x/y)/z = x/(yz) are not equal.

c) Let R under * given by x*y = |xy|.

for x = 0 in R there is no invrse in R , so R is not a group