0
$\begingroup$

Let $G$ be a finite 3-group of maximal class. The center $Z(G)$ contains two elements other than the identity. Does there exist an endomorphism of $G$ that maps one of them to the other?

This is true for the six 3-groups of maximal class of order at most 81. It is also true for the group with GAP ID [729, 46] (though it is not true in this case that there is an automorphism mapping one element to the other as pointed out by LeechLattice) as the following Sagemath code affirms:

g = gap.SmallGroup(729,46);
checker1 = False;        
list1 = gap.AsList(gap.Centre(g));
a = list1[1];
b = list1[2];
list2 = gap.AsList(gap.AllEndomorphisms(g));
for m in list2:
    if not gap.IsInjective(m):
        if gap.Image(m, a) == b or gap.Image(m, b) == a:
            checker1 = True;
print(checker1);
Improve this question
$\endgroup$
2
  • 2
    $\begingroup$ It's not a good idea to change the goal post after the fact; at the very least, please make it clear that you changed it thus (BTW the title still says "automorphism", but the body of your question now asks for endomorphisms). @LeechLattice answered your questions, and even showed you how to do it. Now you changed the question, making their answer looking unsuitable. $\endgroup$
    – Max Horn
    Commented Jan 13, 2020 at 7:06
  • 2
    $\begingroup$ Anyway, have you considered just trying yourself to use GAP (directly or via SageMath) to test your new question? After all, it also contains groups of order 3^6 and 3^7, and with a look at the GAP manual, it shouldn't be too hard to figure out how to find all endomorphisms of a group (= all homomorphisms of the group into itself). Or you could ask for that. $\endgroup$
    – Max Horn
    Commented Jan 13, 2020 at 7:08

1 Answer 1

Reset to default
2
$\begingroup$

The group SmallGroup(729,46) is a counterexample, as verified by GAP calculations:

gap> B:=SmallGroup(729,46);
<pc group of size 729 with 6 generators>
gap> Order(Center(B));
3
gap> CompositionSeries(B);
[ Group([ f1, f2, f3, f4, f5, f6 ]), Group([ f2, f3, f4, f5, f6 ]), Group([ f3, f4, f5, f6 ]), Group([ f4, f5, f6 ]),
  Group([ f5, f6 ]), Group([ f6 ]), Group([  ]) ]
gap> Order(AutomorphismGroup(B));
6561

As $\text{Aut}(B)$ is of odd order, it couldn't contain any element which swaps the non-identity center elements, as such an element is necessarily of even order.

Improve this answer
$\endgroup$
0

You must log in to answer this question.