Forum Problem Lists Definitions FAQ

17.44 (2010)

Solved

Let $\pi$ be a set of primes. A finite group is called a $C_\pi$-group if it possesses exactly one class of conjugate Hall $\pi$-subgroups. A finite group is called a $D_\pi$-group if any two of its maximal $\pi$-subgroups are conjugate.
$\qquad$ a) In a $C_\pi$-group, is an overgroup of a Hall $\pi$-subgroup always a $C_\pi$-group? — An affirmative answer in the case $2 \notin \pi$ follows mod CFSG from (F. Gross, Bull. London Math. Soc., 19, no. 4 (1987), 311–319).
$\qquad$ b) In a $D_\pi$-group, is an overgroup of a Hall $\pi$-subgroup always a $D_\pi$-group?

Contributor: E. P. Vdovin, D. O. Revin

Progress

a) Yes, it is (E. P. Vdovin, D. O. Revin, Siberian Math. J., 54, no. 1 (2013), 22–28).

b) Yes, it is (N. Ch. Manzaeva, https://arxiv.org/pdf/1504.03137.pdf) and (E. P. Vdovin, N. Ch. Manzaeva, D. O. Revin, Sb. Math., 211, no. 3 (2020), 309–335).

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.