Zalesskii group

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Definition

The Zalesskii group can be defined as the following subgroup of unitriangular matrix group:UT(3,Q). Note that the group is not unique up to isomorphism, because the choice of group depends on the choice of prime partitioning in Step (1). The number of such groups up to isomorphism equals the cardinality of the continuum.

Step no. Step detail
1 Partition the set of all prime numbers into three pairwise disjoint subsets L,M,N where 2 \in N, all three of the subsets being infinite.
2 Let A be the set \left \{ a_p = \begin{pmatrix} 1 & 1/p & 1/p^2 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\\end{pmatrix} \mid p \in L \right \}
3 Let B be the set \left \{ b_p = \begin{pmatrix} 1 & 0 & 1/p^2 \\ 0 & 1 & 1/p \\ 0 & 0 & 1 \\\end{pmatrix} \mid p \in M \right \}
4 Let C be the set \left \{ c_p = \begin{pmatrix} 1 & 1/p & 0 \\ 0 & 1 & 1/p \\ 0 & 0 & 1 \\\end{pmatrix} \mid p \in N \right \}
5 The group G we are interested in is the subgroup G = \langle A \cup B \cup C \rangle inside UT(3,\mathbb{Q}), the group of 3 \times 3 upper-triangular matrices with 1s on the diagonal and rational entries.

The group was constructed by Zalesskii to demonstrate that there exist infinite nilpotent groups in which every automorphism is inner.

References

Original construction

  • An example of a torsion-free nilpotent group having no outer automorphisms by A. E. Zalesskii, Matematicheskie Zametki, Volume 11,Number 1, Page 21 - 26(January 1972): More info