# Can a simple group be equivalent to a non-simple group?

Two abstract groups $G$ and $H$ are called equivalent, $G\sim H$, if each of them is isomorphic to a subgroup of another.

Question: Can a simple group $G$ be equivalent to a non-simple group $H$?

Of course, we are talking about infinite groups here. Thanks.

• Yes it's easy, the group $A$ of finitely supported even permutations of the integers and $A\times F$ for any nontrivial finite group.
– YCor
May 13, 2017 at 12:25
Yes. (This is corrected and expanded since the first version.) There are easy examples of simple groups $G$ such that $G\times G$ is isomorphic to a subgroup of $G$. One example is the group of finitely supported even permutations of a countable infinite set. Another is the quotient of all permutations of a countably infinite set by the finitely supported ones. Another is the infinite special linear group (direct limit of $SL_n(k)$) of a field.