TITLE: Is there a set which is a group with respect to both addition and multiplication?
QUESTION [13 upvotes]: Since addition requires 0 as it's identity and 0 has no inverse under multiplication this would seem to suggest that it is impossible but I am unable to prove it or find an example. Perhaps the rules are different enough under complex numbers, quaternions, or octonions to allow such a set to be possible.

REPLY [2 votes]: The set $\{0\}$ forms a trivial group under both of the operations $+$ and $\times$ given their ordinary meaning on the integers.