# Characteristic subgroups of finite 3-groups

- Reference: `kourovka-21-50-characteristic-subgroups-of-finite-3-groups`
- Page: https://theoremdb.org/statements/kourovka-21-50-characteristic-subgroups-of-finite-3-groups
- Record maturity: Reviewed problem

## Problem

Does every finite 3-group T have a nontrivial characteristic subgroup C such that if T is a Sylow 3-subgroup of a finite group G, then T ∩ G′ = T ∩ H ′, where H = NG (C)? Such a characteristic subgroup is known to exist in p-groups for p ⩾ 5 (G. Glauberman, Math. Z., 117 (1970), 46-56), and for p = 3 there are two characteristic subgroups K1, K2 such that T ∩ G′ = (T ∩ H1′ )(S ∩ H2′ ), where Hi = NG (Ki ) (G. Glauberman, J. Algebra, 648 (2024), 62-86). The group S4 shows that no such characteristic subgroups can be found in some Sylow 2-subgroups. This is Kourovka Notebook Problem \(21.50\).

## Status

The reviewed record remains open.

## Work

### Working on this

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## References

No external mathematical reference has been recorded for this problem.