By part of the definition,
two elements in a group can be put together with the group operation to obtain a third element that is also an element of the group.
However, I am wondering if the converse is also true. So the new statement would be:
For every element in the group, it can be written as the result of two non-identity elements of the group using the group operation.
So here we are not considering the element itself with the identity. Is there any counterexample? Thanks.
Best Answer
Yes, there are counterexamples. Apart from the trivial group, there is also the two element group $C_2=\{e,a\}$, where $a^2=e$. In this group, $a$ is the unique non-identity element and $a^2=e$ so $a$ cannot be written as a product of non-identity elements.
These are the only counter-examples. Indeed, let $G$ be group of cardinality at least three and let $g\in G$. We write $g$ as a product of two non-identity elements. If $g=e$, then take $h\neq e$ and we have $g=e=h*h^{-1}$. If $g\neq e$, then there exists $h\in G\setminus\{e,g\}$ and $g=h*(h^{-1}g)$, with neither $h$ nor $h^{-1}g$ being the identity.