Forum Problem Lists Definitions FAQ

12.22 (1992)

Solved

Let $\Delta(G)$ be the augmentation ideal of the integer group ring of an arbitrary group $G$. Then $D_n(G) = G \cap (1 + \Delta^n(G))$ contains the $nth$ lower central subgroup $\gamma_n(G)$ of $G$.
$\qquad$ a) Is it true that $D_n(G)/\gamma_n(G)$ is central in $G/\gamma_n(G)$?
$\qquad$ b) Is it true that $D_n(G)/\gamma_n(G)$ has exponent dividing 2?

Contributor: N. D. Gupta, Yu. V. Kuz’min

Progress

a) No, not always; moreover, $D_n(G)/\gamma_n(G)$ need not be contained in any term of the upper central series of $G/\gamma_n(G)$ with fixed number (N. D. Gupta, Yu. V. Kuz’min, J. Pure Appl. Algebra, 104, no. 1 (1995), 191–197).
b) No, not always (L. Bartholdi, R. Mikhailov, J. Topology, 16, no. 2 (2023), 822–853).

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.