On central endomorphisms of a group

Document Type: Ischia Group Theory 2014

Author

Seconda Universita di Napoli

Abstract

Let $\Gamma$ be a normal subgroup of the full automorphism group $Aut(G)$ of a group $G$‎, ‎and assume that $Inn(G)\leq \Gamma$‎. ‎An endomorphism $\sigma$ of $G$ is said to be $\Gamma$-central if $\sigma$ induces the the identity on the factor group $G/C_G(\Gamma)$‎. ‎Clearly‎, ‎if $\Gamma=Inn(G)$‎, ‎then a $\Gamma$-central endomorphism is a central endomorphism‎. ‎In this article the conditions under which a $\Gamma$-central endomorphism of a group is an automorphism are investigated‎.

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J. E. Adney and T. Yen (1965). Automorphisms of a p-group. Il linois J. Math.. 9, 137-143
H. Dietrich and P. Moravec (2011). On the autocomutator subgroup and absolute centre of a group. J. Algebra. 341, 150-157
G. Endimioni (2004). Hopficity and Co-hopficity in soluble groups. Ukrainian Math. J.. 56, 1594-1601
S. Franciosi, F. de Giovanni and M. L. Newell (1994). On central automorphisms of infinite groups. Comm. Algebra. 22, 2559-2578
F. de Giovanni, M. L. Newell and A. Russo (2014). A note on fixed p oints of automorphisms of infinite groups. Int. J. Group Theory. 3 (4), 57-61
P. Hall (1940). The classification of prime-p ower groups. J. Reine Angew. Math.. 182, 130-141
P. Hegarty (1994). The absolute centre of a group. J. Algebra. 169, 929-935
M. R. R. Moghaddam and H. Safa (2010). Some properties of autocentral automorphisms of a group. Ric. Mat.. 59, 257-264
D. J. S. Robinson (1996). A Course in the theory of groups. Springer-Verlag, New York.