Abstract
A group G is called capable if there is a group H such that \({G \cong H/Z(H)}\) is isomorphic to the group of inner automorphisms of H. We consider the situation that G is a finite capable p-group for some prime p. Suppose G has rank \({d(G) \ge 2}\) and Frattini class \({c \ge 1}\), which by definition is the length of a shortest central series of G with all factors being elementary abelian. There is up to isomorphism a unique largest p-group \({G_d^c}\) with rank d and Frattini class c, and G is an epimorphic image of \({G_d^c}\). We prove that this \({G_d^c}\) is capable; more precisely, we have \({G_d^c \cong G_d^{c+1}/Z(G_d^{c+1})}\).
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References
Beyl F.R., Felgner U., Schmid P.: On groups occurring as center factor groups. J. Algebra 61, 161–177 (1979)
Blackburn N., Evens L.: Schur multipliers of p-groups. J. Reine Angew. Math. 309, 100–113 (1979)
Bryant R.M., Kovács L.G.: Lie representations and groups of prime power order. J. London Math. Soc. (2) 17, 415–421 (1978)
Hall P.: The classification of prime power groups. J. Reine Angew. Math. 182, 130–141 (1940)
Huppert B., Blackburn N.: Finite Groups II. Springer, New York (1982)
Lazard M.: Sur les groupes nilpotents et les anneaux de Lie. Ann. Sci. École Norm. Sup. 71, 101–190 (1954)
I.R. Šafarevič (Shafarevich), On the construction of fields with a given Galois goup of order \({\ell^{a}}\), Amer. Math. Soc. Transl. 4 (1956), 107–142.
Shafarevich I.R.: Factors of a decreasing central series. Math. Notes 54, 262–264 (1989)
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Schmid, P. On a class of finite capable p-groups. Arch. Math. 106, 301–304 (2016). https://doi.org/10.1007/s00013-016-0872-8
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DOI: https://doi.org/10.1007/s00013-016-0872-8