1
$\begingroup$

Can a set with one element be a group, or does a group need to have at least two elements?

$\endgroup$
4
  • $\begingroup$ Try defining a binary operation on a singleton set and see if it is a group. $\endgroup$
    – James
    Sep 15, 2014 at 13:54
  • 1
    $\begingroup$ The number $1$ is not a group. A set with a single element can be made a group, however. $\endgroup$ Sep 15, 2014 at 13:57
  • $\begingroup$ Thanks for the answer, Thomas Andrews. Can you explain how a single element can be made a group? By adding to it? $\endgroup$
    – user176203
    Sep 15, 2014 at 14:09
  • $\begingroup$ “In mathematics, a trivial group is a group consisting of a single element”. $\endgroup$
    – MJD
    Sep 15, 2014 at 18:00

1 Answer 1

5
$\begingroup$

It is possible to have an element with just one element. Take $G=\{e\}$ with the group law defined as $e\circ e=e$. Then it is easy to verify that:

  1. The group law is associative: $(e\circ e)\circ e=e\circ(e\circ e)=e$
  2. There is an identity $e$ such that $e\circ g=g = g \circ e$ for all $g\in G$.
  3. Every element of $G$ has an inverse, since $e$ is its own inverse.
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .