Inertial endomorphisms of an abelian group
Tóm tắt
We describe inertial endomorphisms of an abelian group
$$A$$
, that is endomorphisms
$$\varphi $$
with the property
$$|(\varphi (X)+X)/X|<\infty $$
for each
$$X\le A$$
. They form a ring
$$IE(A)$$
containing the ideal
$$F(A)$$
formed by the so-called finitary endomorphisms, the ring of power endomorphisms and also other non-trivial instances. We show that the quotient ring
$$IE(A)/F(A)$$
is commutative. Moreover, inertial invertible endomorphisms form a group, provided
$$A$$
has finite torsion-free rank. In any case, the group
$$IAut(A)$$
they generate is commutative modulo the group
$$FAut(A)$$
of finitary automorphisms, which is known to be locally finite. We deduce that
$$IAut(A)$$
is locally-(center-by-finite). Also, we consider the lattice dual property, that is
$$|X/(X\cap \varphi (X))|<\infty $$
for each
$$X\le A$$
and show that this implies the above one, provided
$$A$$
has finite torsion-free rank.
Tài liệu tham khảo
Belyaev, V.V., Kuzucuoglu, M., Seckin, E.: Totally inert groups. Rend. Semin. Mat. Univ. Padova 102, 151–156 (1999)
Casolo, C.: Groups with finite conjugacy classes of subnormal subgroups. Rend. Semin. Mat. Univ. Padova 81, 107–149 (1989)
Dardano, U., Rinauro, S.: Inertial automorphisms of an abelian group. Rend. Semin. Mat. Univ. Padova 127, 213–233 (2012)
Dardano, U., Rinauro, S.: On the ring of inertial endomorphisms of an abelian group. Ricerche Mat. (2014). doi:10.1007/s11587-014-0199-3
De Falco, M., de Giovanni, F., Musella, C., Trabelsi, N.: Strongly inertial groups. Commun. Algebra 41, 2213–2227 (2013)
Dikranjan, D., Giordano Bruno, A., Salce, L., Virili, S.: Fully inert subgroups of divisible Abelian groups. J. Group Theory 16, 915–939 (2013)
Dikranjan, D., Giordano Bruno, A., Salce, L., Virili, S.: Intrinsic algebraic entropy. J. Pure Appl. Algebra (2014). doi:10.1016/j.jpaa.2014.09.033
Dixon, M., Evans, M.J., Tortora, A.: On totally inert simple groups. Cent. Eur. J. Math. 8(1), 22–25 (2010)
Franciosi, S., de Giovanni, F., Newell, M.L.: Groups whose subnormal subgroups are normal-by-finite. Commun. Alg. 23(14), 5483–5497 (1995)
Fuchs, L.: Infinite Abelian Groups. Academic Press, New York (1970–1973)
Robinson, D.J.S.: On inert subgroups of a group. Rend. Semin. Mat. Univ. Padova 115, 137–159 (2006)
Specht, W., Heineken, H.: Gruppen mit endlicher Komponentenzahl fastgleicher Untergruppen. Math. Nachr. 134, 73–82 (1987)
Wehrfritz, B.A.F.: Finite-finitary groups of automorphisms. J. Algebra Appl. 1(4), 375–389 (2002)