Forum Problem Lists Definitions FAQ

8.3 (1982)

Open

Let $m$ and $n$ be positive integers and $p$ a prime. Let $P_n(\mathbb{Z}_{p^m})$ be the group of all $n \times n$ matrices $(a_{ij})$ over the integers modulo $p^m$ such that $a_{ii} \equiv 1$ for all $i$ and $a_{ij} \equiv 0 \pmod{p}$ for all $i > j$. The group $P_n(\mathbb{Z}_{p^m})$ is a finite $p$-group. For which $m$ and $n$ is it regular?

Contributor: B. A. F. Wehrfritz

Progress

Editors’ comment (2001): The group $P_n(\mathbb{Z}_{p^m})$ is known to be regular if $mn < p$ (Yu. I. Merzlyakov, Algebra i Logika, 3, no. 4 (1964), 49–59 (Russian)). The case $m = 1$ is completely done by A. V. Yagzhev in (Math. Notes, 56, no. 5–6 (1995), 1283–1290), although this paper is erroneous for $m > 1$. The case $m = 2$ is completed in (S. G. Kolesnikov, Issledovaniya in analysis and algebra, no. 3, Tomsk Univ., Tomsk, 2001, 117–125 (Russian)).

Editors’ comment (2009): The group $P_n(\mathbb{Z}_{p^m})$ was shown to be regular for any $m$ if $n^2 < p$ (S. G. Kolesnikov, Siberian Math. J., 47, no. 6 (2006), 1054–1059).

Comments

All comments are the responsibility of the user. Comments appearing on this page are not verified for correctness. Please keep posts mathematical and on topic. If you want to submit a proof (or a partial proof), please use the dedicated proof submission form rather than posting it in the comments.
Order by newest first or oldest first.

No comments yet. Be the first to comment.

Proof claims

Proof claims are the responsibility of the submitter. Appearance here does not mean the claim has been checked for mathematical correctness. Moderators only screen for spam, abuse, and obviously low-effort submissions.

No proof claims yet.