# Characteristic subgroups of a finite abelian $2$-group

I have recently stumbled across the problem of describing the characteristic subgroups of a finite abelian group. With some discussions with some mathematicians in my lab, I managed to obtain a "visual" description of such, and wrote it down.

Nonetheless, I know that some of this work has already been clear by Kaplansky in "Infinite abelian groups". Combining both theorems 25 and 26, we obtain a description for every finite abelian $$p$$-group for $$p\neq 2$$, and then for every abelian finite group of odd order. I managed to obtain a (less aesthetical, but still understoodable) description for the finite abelian $$2$$-groups that Kaplansky mentions in the theorem 27 as problematic. In this book, I didn't manage to find some reference for such groups, but maybe some person here knows an article that has already tackles this question.

So my question is : has some text already given a reasonable parametrization of characteristic subgroups of finite abelian $$2$$-groups ?

• Could you say something more about what your visual description looks like? Commented Jan 16, 2023 at 16:46
• Kaplansky associate in substance somme function to any element and have a simple description of wich set of functions will give a fully invariant subgroup. I juste help myself from the visualization of the graph of this function and try to describe with few information the set of graphs that would give a characteristic subgroup. Commented Jan 16, 2023 at 16:59
• It seems to me that there is the requisite description here Kerby, B. L., Rode, E. Characteristic Subgroups of Finite Abelian Groups. Communications in Algebra, 2011, 39:4, 1315-1343 (Section 2). Commented Jan 17, 2023 at 11:58
• Thanks for this reference, which is what I looked for ! Commented Jan 18, 2023 at 9:21
• @kabenyuk you can maybe post it as answer so that I can mark it down. Commented Jan 18, 2023 at 9:29