7
$\begingroup$

This is related to If $G$ and $H$ are nonisomorphic group with same order then can we say that $Aut(G)$ is not isomorphic to $Aut(H)$? and Can non-isomorphic abelian groups have isomorphic endomorphism rings? but more general than both. The answer given in the first link says that two finite non-isomorphic groups can have isomorphic automorphism groups. The second link (apparently) gives an example of two non-isomorphic infinite groups with isomorphic endomorphism rings. But if $G$ and $H$ are two non-isomorphic finite groups of the same order, is it possible for their endomorphism monoids to be isomorphic?

$\endgroup$
4
$\begingroup$

Not necessarily.

I cannot think of a simple example right now, but there are plenty of examples of finite $p$-groups $G$ of nilpotence class $2$ such that

  1. all automorphisms are central, and
  2. an endomorphism that is not an automorphism maps $G$ into $Z(G)$.

Now if you take two non-isomorphic such groups $G_{1}, G_{2}$ in which the $G_{i}/Z(G_{i})$ and the $Z(G_{i})$ are elementary abelian and isomorphic, then their endomorphism monoid are isomorphic.

As a reference, see this paper of mine (should be freely accessible, in case please advise), particularly Section 2 and Theorem 4.3.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.