Many of people have tried to obtain the structure of Baer-invariant of groups exactly. Recently, B. Mashayekhy and M. Parvizi determine Baer-invariant of finitely generated Abelian groups. Also, it is done for some no...Many of people have tried to obtain the structure of Baer-invariant of groups exactly. Recently, B. Mashayekhy and M. Parvizi determine Baer-invariant of finitely generated Abelian groups. Also, it is done for some non-Abelian groups, such the dihedral and the quaternion groups directly, and sometimes with the softwares of Gap and Magma. But nobody works on non finitely generated Abelian groups. In 1979, M.R.R. Moghaddarn showed that the structure of Baer-invariant of group commutes with the direct limit of a directed system, in some sense. The authors have used these results and proved that the Baer-invariant of C is always trivial and also Baer-invariant of Abelian groups Q/z and Z （p∞）, with respect to the varieties of outer commutators and so polynilpotent, nilpotent are trivial. One can see immediately that the covering groups of these groups are themselves. Then after computing the Baer-invariant of Zn with respect to Burnside variety, we have concluded for Q/z and Z（P∞） Burnside variety. In the future, they try to survey the commutativity of the Baer-invariant variety with the other useful varieties in order to attain similar results for another non finitely generated Abelian groups.展开更多
Monomorphism categories of the symmetric and alternating groups are studied via Cayley's Em-bedding Theorem. It is shown that the parity is well defined in such categories. As an application, the parity in a finit...Monomorphism categories of the symmetric and alternating groups are studied via Cayley's Em-bedding Theorem. It is shown that the parity is well defined in such categories. As an application, the parity in a finite group G is classified. It is proved that any element in a group of odd order is always even and such a group can be embedded into some alternating group instead of some symmetric group in the Cayley's theorem. It is also proved that the parity in an abelian group of even order is always balanced and the parity in an nonabelian group is independent of its order.展开更多
文摘Many of people have tried to obtain the structure of Baer-invariant of groups exactly. Recently, B. Mashayekhy and M. Parvizi determine Baer-invariant of finitely generated Abelian groups. Also, it is done for some non-Abelian groups, such the dihedral and the quaternion groups directly, and sometimes with the softwares of Gap and Magma. But nobody works on non finitely generated Abelian groups. In 1979, M.R.R. Moghaddarn showed that the structure of Baer-invariant of group commutes with the direct limit of a directed system, in some sense. The authors have used these results and proved that the Baer-invariant of C is always trivial and also Baer-invariant of Abelian groups Q/z and Z （p∞）, with respect to the varieties of outer commutators and so polynilpotent, nilpotent are trivial. One can see immediately that the covering groups of these groups are themselves. Then after computing the Baer-invariant of Zn with respect to Burnside variety, we have concluded for Q/z and Z（P∞） Burnside variety. In the future, they try to survey the commutativity of the Baer-invariant variety with the other useful varieties in order to attain similar results for another non finitely generated Abelian groups.
文摘Monomorphism categories of the symmetric and alternating groups are studied via Cayley's Em-bedding Theorem. It is shown that the parity is well defined in such categories. As an application, the parity in a finite group G is classified. It is proved that any element in a group of odd order is always even and such a group can be embedded into some alternating group instead of some symmetric group in the Cayley's theorem. It is also proved that the parity in an abelian group of even order is always balanced and the parity in an nonabelian group is independent of its order.
