A month ago, I took a test and was asked whether “a group of 689 elements is closed under the group operation”. The question is trivial, but one can discover another question: Does such a group exists?

In order to “proceed with with my work”, I temporarily accepted the existence fo swuch a group during the test. However, even though I got the question right and earned some marks for that, I didn’t think that this is true mathematics. To say that one knows this quoted statement, one has to follow the “international standard” of a group and make rigourous arguements from the “basic laws” to prove the existence of such a group.

Thinking that the group of 689 elements isn’t useful, I put that problem aside. After a fortnight, I came up with an answer: $\Z_{13} \times \Z_{53} \cong \Z_{689}$.