Finite 2-groups in which the set of self-centralizing abelian normal subgroups with at least three generators is empty ($SCN_3(2)=\varnothing$)
Izvestiya. Mathematics , Tome 7 (1973) no. 2, pp. 247-280.

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In this paper we study finite 2-groups in which each abelian normal subgroup is metacyclic, i.e. $SCN_3(2)=\varnothing$. The main result: a finite 2-group with $SCN_3(2)=\varnothing$ is an extension of a metacyclic group by a group isomorphic to a subgroup of the dihedral group of order 8.
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A. D. Ustyuzhaninov. Finite 2-groups in which the set of self-centralizing abelian normal subgroups with at least three generators is empty ($SCN_3(2)=\varnothing$). Izvestiya. Mathematics , Tome 7 (1973) no. 2, pp. 247-280. http://geodesic.mathdoc.fr/item/IM2_1973_7_2_a0/

[1] Blackburn N., “Generalization of certain elementary theorems on $p$-groups”, Proc. London Math. Soc., XI:41 (1961), 1–22 | DOI | MR

[2] Feit W. and Thompson J. G., “Solvability of groups of odd order”, Pacif. J. Math., 13:3 (1963), 775–1029 | MR | Zbl

[3] Thompson J. G., “Nonsolvable finite groups all of whose local subgroups are solvable”, Bull. Amer. Math. Soc., 74:3 (1968), 383–428 | DOI | MR

[4] Alperin J. L., “Centralizers of Abelian Normal Subgroups of $p$-groups”, Z. Algebra, 1 (1964), 110–113 | DOI | MR | Zbl

[5] Ustyuzhaninov A. D., “Konechnye 2-gruppy s tremya involyutsiyami”, Sib. matem. zh., XIII:1 (1972), 182–197

[6] Mac. Williams A. R., “On 2-groups with no normal abelian subgroups of rank 3, and their occurence as Sylow 2-subgroups of finite simple groups”, Trans. Amer. Math. Soc., 150:2 (1970), 345–408 | DOI | MR | Zbl

[7] Konvisser M. W., “A class of 2-groups”, Notic Amer. Math. Soc., 18:1 (1971), 128, Preliminary report

[8] Ustyuzhaninov A. D., “O konechnykh $p$-gruppakh, kommutant kazhdoi sobstvennoi podgruppy kotorykh metatsiklicheskii”, Sib. matem. zh., 3 (1971), 834–844

[9] Bechtel H., “Frattini subgroups an $\Phi$-central groups”, Pacif. J. Math., 18:1 (1966), 15–23 | MR | Zbl

[10] Ustyuzhaninov A. D., “Konechnye gruppy s invariantnymi netsiklicheskimi podgruppami”, Matem. zapiski, 6:1 (1967), 107–123 | MR | Zbl

[11] Kholl M., Teoriya grupp, IL, M., 1962

[12] Suzuki M., “On finite groups containing on element of order four which commutes only with its power”, Illinois J. Math., 3:2 (1955), 255–271 | MR

[13] Busarkin V. M., Starostin A. I., “Konechnye gruppy, vse sobstvennye podgruppy kotorykh obladayut nilpotentnym rasschepleniem”, Izv. AN SSSR. Ser. matem., 29 (1965), 97–108 | MR | Zbl

[14] Sheriev V. A., “Konechnye 2-gruppy s dopolnyaemymi neinvariantnymi podgruppami”, Sib. matem. zh., 7:1 (1967), 266–315

[15] Redei L., “Das schiefe Produkt in der Gruppen theorie mit Anwendungen auf die endlichen nicht kommutativen echten Untergruppen und die Ordnung szahlen, zu denen nur kommutativen Gruppen gehören”, Comm. Math. Helv., 20 (1947), 252–264 | DOI | MR

[16] Schur J., “Untersuchungen über die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen”, J. für Math., 132 (1907), 85–137 | Zbl