Further research problems and theorems on prime power groups. (English) Zbl 1478.20013
From the authors’ summary: “A great number of research problems concerning finite \(p\)-groups are stated. This list is a continuation of the six lists in [the authors, Groups of prime power order. Vol. 1. Berlin: Walter de Gruyter (2008; Zbl 1168.20001); Vol. 2 (2008; Zbl 1168.20002); Vol. 3 (2011; Zbl 1229.20001); Vol. 4 (2016; Zbl 1344.20001); Vol. 5 (2016; Zbl 1344.20002); Vol. 6 (2018; Zbl 1400.20001]. Also stated are new theorems with proofs.”
Reviewer’s remarks: In Volume 1 [Zbl 1168.20001] of the series of books “Groups of prime power order” by the first author, and in Volumes 2 up to and including 6 [Zbl 1168.20002; Zbl 1229.20001; 1344.20001; Zbl 1344.20002; Zbl 1400.20001], both authors describe extensive theorems, research problems, and unsolved problems, all dealing indeed with \(p\)-groups of finite order. In this paper, one finds an update of unsolved problems as well as new research problems, nine-hundred items (!) numbered 4001–5000. Some of these have been solved, and mentioned with their proofs, see 4709, 4775, 4815, 4844 up to and including 4848, 4893 and 4919. All their books and the paper under review are an immense toolkit in studying finite \(p\)-groups.
Reviewer’s remarks: In Volume 1 [Zbl 1168.20001] of the series of books “Groups of prime power order” by the first author, and in Volumes 2 up to and including 6 [Zbl 1168.20002; Zbl 1229.20001; 1344.20001; Zbl 1344.20002; Zbl 1400.20001], both authors describe extensive theorems, research problems, and unsolved problems, all dealing indeed with \(p\)-groups of finite order. In this paper, one finds an update of unsolved problems as well as new research problems, nine-hundred items (!) numbered 4001–5000. Some of these have been solved, and mentioned with their proofs, see 4709, 4775, 4815, 4844 up to and including 4848, 4893 and 4919. All their books and the paper under review are an immense toolkit in studying finite \(p\)-groups.
Reviewer: Robert W. van der Waall (Huizen)
Keywords:
lists of open problems; \(p\)-groups of exponent \(p\); metacyclic \(p\)-groups; metahamiltonian \(p\)-groups; self-centralizing subgroups; \(H_p\)-subgroups; maximal abelian subgroups; minimal nonabelian subgroups; regular \(p\)-groupsCitations:
Zbl 1168.20001; Zbl 1168.20002; Zbl 1229.20001; Zbl 1344.20001; Zbl 1344.20002; Zbl 1400.20001References:
| [1] | [B1]Y. Berkovich, Groups of Prime Power Order, Vol. 1, Walter de Gruyter, Berlin, 2008. · Zbl 1168.20002 |
| [2] | [BJ2] Y. Berkovich and Z. Janko, Groups of prime power order, Vol. 2, Walter de Gruyter, Berlin, 2008. · Zbl 1168.20002 |
| [3] | [BJ3] Y. Berkovich and Z. Janko, Groups of prime power order, Vol. 3, Walter de Gruyter, Berlin, 2011. · Zbl 1229.20001 |
| [4] | [BJ4] Y. Berkovich and Z. Janko, Groups of prime power order, Vol. 4, Walter de Gruyter, Berlin, 2016. · Zbl 1344.20002 |
| [5] | [BJ5] Y. Berkovich and Z. Janko, Groups of prime power order, Vol. 5, Walter de Gruyter, Berlin, 2016. · Zbl 1344.20001 |
| [6] | [BJ6] Y. Berkovich and Z. Janko, Groups of prime power order, Vol. 6, Walter de Gruyter, Berlin, 2018. · Zbl 1400.20001 |
| [7] | [Cos]J. Cossey, Special maximal subgroups of p-groups, Bull. Aust. Math. Soc. 89 (2014), 415-419. · Zbl 1297.20015 |
| [8] | [Hor1] M.W. Horosevskii, The automorphism groups of finite p-groups, Algebra i Logika 10(1971), 81-86 (in Russian). |
| [9] | [Hor2] M.W. Horosevskii, Automorphisms of finite groups, Mat. Sb. (N.S.) 93 (1974), 576- 587 (in Russian). · Zbl 0291.20029 |
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