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

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 ?

• There is more than one set of axioms for group theory. Your talk of "left axioms" implies that you're using a less common one than average. Please would you let us know, too, how you got your current results. You could be closer than you think! Sep 25 '19 at 23:41
• @Shaun sorry sir, but I can't find my answers there. I can't understand what that is written in that post, most of them are in codings. Sep 25 '19 at 23:41
• My link above is to a meta site question on how to use $\LaTeX$ to format mathematics on the main site. Sep 25 '19 at 23:43
• @Shaun please see this ez=|e|z, here the identity element e can be any member of the set. But if ze=|z|e , then identity element can be only z itself. In first case the identity element is left identity and in second case is right identity. Sep 25 '19 at 23:45
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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.