The automorphism group for $p$-central $p$-groups

Document Type : Research Paper


University of Cambridge, UK


A $p$-group $G$ is $p$-central if $G^{p}\le Z(G)$‎, ‎and $G$ is‎ ‎$p^{2}$-abelian if $(xy)^{p^{2}}=x^{p^{2}}y^{p^{2}}$ for all $x,y\in‎ ‎G$‎. ‎We prove that for $G$ a finite $p^{2}$-abelian $p$-central‎ ‎$p$-group‎, ‎excluding certain cases‎, ‎the order of $G$ divides the‎ ‎order of $\text{Aut}(G)$‎.


Main Subjects

J. Buckley (1975). Automorphism groups of isoclinic $p$-groups. J. London Math. Soc. (2). 12, 37-44 R. M. Davitt (1972). The automorphism group of finite $p$-abelian $p$-groups. Illinois J. Math.. 16, 76-85 R. M. Davitt (1980). On the automorphism group of a finite $p$-group with a small central quotient. Canad. J. Math.. 32, 1168-1176 R. M. Davitt & A. D. Otto (1972). On the automorphism group of a finite modular $p$-group. Proc. Amer. Math. Soc.. 35 (2), 399-404 R. M. Davitt & A. D. Otto (1971). On the automorphism group of a finite $p$-group with the central quotient metacyclic. Proc. Amer. Math. Soc.. 30 (3), 467-472 T. Exarchakos (1989). On $p$-groups of small order. Publ. Inst. Math. (Beograd) (N. S.) (59). 45, 73-76 R. Faudree (1968). A note on the automorphism group of a $p$-group. Proc. Amer. Math. Soc.. 19, 1379-1382 S. Fouladi, A. R. Jamali & R. Orfi (2007). Automorphism groups of finite $p$-groups of coclass 2. J. Group Theory. 10 (4), 437-440 W. Gasch"{u}tz (1966). Nichtabelsche $p$-Gruppen besitzen "{a}ussere $p$-Automorphismen (German). J. Algebra. 4, 1-2 N. Gavioli (1993). The number of automorphisms of groups of order $p^{7}$. Proc. Roy. Irish Acad. Sect.. 93 (2), 177-184 C. J. Hillar & D. L. Rhea (2007). Automorphisms of finite abelian groups. Amer. Math. Monthly. 114 (10), 917-923 K. G. Hummel (1975). The order of the automorphism group of a central product. Proc. Amer. Math. Soc.. 47 (1), 37-40 C. R. Leedham-Green & S. McKay (2002). The Structure of Groups of Prime Power Order. London Mathematical Society Monographs New Series, Oxford Science Publications. 27 A. D. Otto (1966). Central automorphisms of a finite $p$-group. Trans. Amer. Math. Soc.. 125, 280-287 I. B. S. Passi, M. Singh & M. K. Yadav (2010). Automorphisms of abelian group extensions. J. Algebra (4). 324, 820-830 A. Ranum (1907). The group of classes of congruent matrices with application to the group of isomorphisms of any abelian group. Trans. Amer. Math. Soc.. 8, 71-91 A. Thillaisundaram (2011). PhD Thesis. University of Cambridge, UK. M. K. Yadav (2007). On automorphisms of finite $p$-groups. J. Group Theory. 10 (6), 859-866