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Test ranks of finitely generated Abelian groups. (English) Zbl 1013.20049
Cleary, Sean (ed.) et al., Combinatorial and geometric group theory. Proceedings of the AMS special session on combinatorial group theory, New York, NY, USA, November 4-5, 2000 and the AMS special session on computational group theory, Hoboken, NJ, USA, April 28-29, 2001. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 296, 199-206 (2002).
The following problem is in the Magnus problem list at http://zebra.sci.ccny.cuny.edu/ web/problems: (FP15) (B. Fine) “Let $$G$$ be an $$n$$-generator group. Call a set of elements $$\{g_1,\dots,g_k\}$$, $$k\leq n$$, a ‘test set’ for the group $$G$$ if, whenever $$f(g_i)=g_i$$, $$i=1,\dots,k$$, for some endomorphism $$f$$ of the group $$G$$, this $$f$$ is actually an automorphism of $$G$$. The ‘test rank’ of $$G$$ is the minimal cardinality of a test set. Can the test rank be equal to 2 if $$n>2$$?”
In 2000 Timoshenko answered this question in the affirmative by showing that the free metabelian group of rank 3 has test rank 2. In this paper, the authors give a more thorough answer to the above question by showing that for any pair of integers $$k$$ and $$n$$ with $$1\leq k\leq n$$ there are finite Abelian $$p$$-groups of rank $$n$$ and test rank $$k$$. This is accomplished by calculating the test rank of such a group as follows:
Theorem: Suppose that $$A$$ is a finite Abelian $$p$$-group, with canonical decomposition $$A=(\mathbb{Z}_{p^{k_1}})^{r_1}\oplus(\mathbb{Z}_{p^{k_2}})^{r_2}\oplus\cdots\oplus(\mathbb{Z}_{p^{k_n}})^{r_n}$$, $$k_i<k_{i+1}$$ and $$1\leq r_i$$ for all $$i$$, and let $$E$$ be the set of integers $$E=\{r_j\mid 1\leq j\leq n\}\cup\{r_j+r_{j+1}\mid 1\leq j<n,\;k_{j+1}=k_j+1\}$$. Then $$\text{test rank}(A)=\max(E)$$.
Thus for integers $$1\leq k\leq n$$, $$G=(\mathbb{Z}_2)^k\oplus\mathbb{Z}_{2^8}\oplus\cdots\oplus\mathbb{Z}_{2^{2(n-k)+1}}$$ is a group with rank $$n$$ and test rank $$k$$.
The authors also generalize the above theorem to compute the test rank of finitely generated Abelian groups.
For the entire collection see [Zbl 0990.00044].
##### MSC:
 20K01 Finite abelian groups 20K30 Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups 20E36 Automorphisms of infinite groups