Group is not Empty

Theorem

A group cannot be empty.


Proof

A group is defined as a monoid for which every element has an inverse.


Thus, as a group is already a monoid, it must at least have an identity, therefore can not be empty.

$\blacksquare$


Sources

  • 1968: Ian D. Macdonald: The Theory of Groups ... (previous) ... (next): $\S 1$: Some examples of groups
This article is issued from ProofWiki. The text is available under Creative Commons Attribution-ShareAlike License unless otherwise noted. Additional terms may apply for the media files.