A class of finite groups with Abelian centralizer of an element of order~$3$ of type $(3,2,2)$
Fundamentalʹnaâ i prikladnaâ matematika, Tome 18 (2013) no. 3, pp. 117-137.

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In this work, we study the structure of finite groups in which the centralizer of an element of order $3$ is isomorphic to $\mathbb Z_3\times\mathbb Z_2\times\mathbb Z_2$. The analysis is restricted to the class of groups whose order is not divisible by the prime number $5$. It is shown that among finite simple groups such groups do not exist, and a detailed possible internal general structure of such groups is investigated. We use only those results that have been published before 1980.
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V. I. Loginov. A class of finite groups with Abelian centralizer of an element of order~$3$ of type $(3,2,2)$. Fundamentalʹnaâ i prikladnaâ matematika, Tome 18 (2013) no. 3, pp. 117-137. http://geodesic.mathdoc.fr/item/FPM_2013_18_3_a7/

[1] Kabanov V. V., Starostin A. I., “Konechnye gruppy, v kotorykh silovskaya $2$-podgruppa tsentralizatora nekotoroi involyutsii poryadka 16”, Mat. zametki, 18:6 (1975), 869–876 | MR | Zbl

[2] Sitnikov V. M., “Konechnye gruppy s silovskoi $2$-podgruppoi, soderzhaschei samotsentralizuyuschuyusya elementarnuyu abelevu podgruppu poryadka 8”, Mat. zametki, 16:6 (1974), 899–906 | MR | Zbl

[3] Sitnikov V. M., “O gruppe Mate $M_{12}$”, Mat. zametki, 15:4 (1974), 651–660 | MR | Zbl

[4] Alperin J. L., Brauer R., Gorenstein D., “Finite groups with quasi-dihedral and wreathed Sylow $2$-subgroups”, Trans. Am. Math. Soc., 151:1 (1970), 1–261 | MR | Zbl

[5] Aschbacher M., “Finite groups with a proper $2$-generated core”, Trans. Am. Math. Soc., 197 (1974), 87–112 | MR | Zbl

[6] Aschbacher M., “Thin finite simple groups”, J. Algebra, 54:1 (1978), 50–152 | DOI | MR | Zbl

[7] Beisiegel B., “Über einfache endliche Gruppen mit Sylow-$2$-Gruppen der Ordnung höchstens $2^{10}$”, Commun. Algebra, 5:2 (1977), 113–170 | DOI | MR | Zbl

[8] Bender H., “Transitive Gruppen gerader Ordnung, in denen jede Involution genau einen Punkt festläßt”, J. Algebra, 17:4 (1971), 527–554 | DOI | MR | Zbl

[9] Brauer R., “Some applications of the theory of blocks of characters of finite groups. III”, J. Algebra, 3:2 (1966), 225–255 | DOI | MR | Zbl

[10] Dickson N. K., “Groups with dihedral $3$-normalizers of order $4k$. I”, J. Algebra, 54:2 (1978), 390–409 | DOI | MR | Zbl

[11] Dickson N. K., “Groups with dihedral $3$-normalizers of order $4k$. II”, J. Algebra, 54:2 (1978), 410–443 | DOI | MR | Zbl

[12] Dickson N. K., “Structure theorems for groups with dihedral $3$-normalisers”, Proc. Edinburgh Math. Soc., 21:2 (1978), 175–186 | DOI | MR | Zbl

[13] Dickson N. K., Page D. R., “Groups with dihedral $3$-normalizers of order $4k$. III”, J. Algebra, 58:2 (1979), 462–480 | DOI | MR | Zbl

[14] Feit W., Thompson J. G., “Finite groups which contain a self-centralizing subgroup of order 3”, Nagoya Math. J., 21, Dec. (1962), 185–197 | MR | Zbl

[15] Glauberman G., Factorizations in Local Subgroups of Finite Groups, Reg. Conf. Ser. Math., 33, Amer. Math. Soc., Providence, 1977 | MR | Zbl

[16] Goldschmidt D. M., “$2$-signalizer functors on finite groups”, J. Algebra, 21:2 (1972), 321–340 | DOI | MR | Zbl

[17] Gorenstein D., Finite Groups, Harper and Row, New York, 1968 | MR | Zbl

[18] Gorenstein D., “On finite simple groups of characteristic 2 type”, Inst. Hautes Études Sci., 36 (1969), 5–13 | DOI | MR | Zbl

[19] Gorenstein D., “The classification of finite simple groups. I. Simple groups and local analysis”, Bull. Am. Math. Soc. (New Ser.), 1:1 (1979), 43–199 | DOI | MR | Zbl

[20] Gorenstein D., Harada K., Finite groups whose $2$-subgroups are generated by at most 4 elements, Mem. Am. Math. Soc., 147, 1974 | MR | Zbl

[21] Gorenstein D., Lyons R., “Nonsolvable finite groups with solvable $2$-local subgroups”, J. Algebra, 38:2 (1976), 453–522 | DOI | MR | Zbl

[22] Gorenstein D., Walter J. H., “Centralizers of involutions in balanced groups”, J. Algebra, 20:2 (1972), 284–319 | DOI | MR | Zbl

[23] Harada K., “On finite groups having self-centralizing $2$-subgroups of small order”, J. Algebra, 33:1 (1975), 144–160 | DOI | MR | Zbl

[24] Morini C., “Alcune osservazioni su una classe di gruppi finiti contenenti un sottogruppo di ordine 3 con prescritto centralizzante. I”, Ann. Univ. Ferrara Sez. VII (N.S.), 23 (1977), 189–194 | MR

[25] Rickman B., “Groups which admit a fixed-point-free automorphism of order $p^2$”, J. Algebra, 59:1 (1979), 77–171 | DOI | MR | Zbl

[26] Stroth G., “Endliche einfache Gruppen mit einer zentralisatorgleichen elementar abelschen Untergruppe von der Ordnung 16”, J. Algebra, 47:2 (1977), 480–523 | DOI | MR | Zbl

[27] Stroth G., “Gruppen mit kleinen $2$-lokalen Untergruppen”, J. Algebra, 47:2 (1977), 441–454 | DOI | MR | Zbl

[28] Wielandt H., “Sylowgruppen und Kompositions-Struktur”, Abh. Math. Sem. Univ. Hamburg, 22:3–4 (1958), 215–228 | DOI | MR | Zbl