I have a doubt regarding the group {4,8,12,16}under multiplication modulo 20. I faced a question show that {4,8,12,16} is a group under multiplication mod 20.It is fine.I have solved the problem also.But I am feeling something strange in it.The idenity element of multiplication modulo 20 should be 1 which is not in the group.I mean does the identity element change it I change the set I am dealing with keeping the binary operation same.Please give me a similiar example in which identity element under the same operation changes due to changing the set in a group.

Set $S=\{4,8,12,16\}$
The identity element of algebraic structure $(S,\times_{20})$ is $16$.



*

*Is this group closed under modulo multiplication: YES

*Is the operation associative: $(4\times_{20}8)\times_{20}12=12\times_{20}12=\textbf{4}=4\times_{20}(8\times_{20}12)=4\times_{20}16=\bf{4}$

*Is there exists identity : YES (16)

*Does each element has its own inverse : YES ($4 = 4^{-1}, 8 = 12^{-1}, 12= 8^{-1}, 16 = 16^{-1}$)


So, this algebraic structure is indeed a group under multiplication modulo 20.
