Show that the set of complex numbers $z$ with $|z|=1$ is not a group under the operation $*$ denoted by $z_1 * z_2 = |z_1|\times z_2.$

Question

Show that the set of complex numbers $z$ with $|z|=1$ is not a group under the operation $*$ denoted by $z_1 * z_2 = |z_1|\times z_2.$

By solving this I found many left identity elements and couldn't find an unique right identitiy. But can't find a particular identity element. Is it enough to say that the set is not a group ?

Also please tell me If a set is satisfying left axioms, but not satisfying right axioms, then we will call the set a group or not ? How many identity element can be in a group ?

Answer 1

Since $\lvert z\rvert=1$ for all $z$ in your set, we have $z_1\ast z_2=z_2$ for all $z_1, z_2$ in the set. This violates the Latin square property of group multiplication. Hence it is not a group.

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Semigroup theory might be of interest to you.