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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 ?

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  • $\begingroup$ Could you say something more about what your visual description looks like? $\endgroup$
    – LSpice
    Commented Jan 16, 2023 at 16:46
  • $\begingroup$ 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. $\endgroup$ Commented Jan 16, 2023 at 16:59
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    $\begingroup$ 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). $\endgroup$
    – kabenyuk
    Commented Jan 17, 2023 at 11:58
  • $\begingroup$ Thanks for this reference, which is what I looked for ! $\endgroup$ Commented Jan 18, 2023 at 9:21
  • $\begingroup$ @kabenyuk you can maybe post it as answer so that I can mark it down. $\endgroup$ Commented Jan 18, 2023 at 9:29

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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).

Theorem 2.9 describes characteristic subgroups of abelian 2-groups.

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