cokernel in Grp

1. Definition / Proposition

Given category Group, groups 20230802-cokernel_of_a_group_bcb6ae55968311c761cdd2456059895470fc3796.svg and 20230802-cokernel_of_a_group_9a1a4fb214bbba01158d81792bccf9c885699d6a.svg a homomorphism. Then the cokernel of 20230802-cokernel_of_a_group_cfc36dd7862e3abe38414721421b74431e668ba7.svg is given by

20230802-cokernel_of_a_group_2ca85894b981b9557df880259b92f843f7a8ccda.svg

where 20230802-cokernel_of_a_group_7b7a2c1532f4e750d9b9074750c902ea486893be.svg is the normal closure

2. Proof

Suppose there exists a group-homomorphism

20230802-cokernel_of_a_group_6de8559ad13a7b566f12a389dac33a33044387af.svg

such that

20230802-cokernel_of_a_group_39cd1954c1bfb8b9de16fb072d57044d3eb4346d.svg

commutes.

Then define

20230802-cokernel_of_a_group_73b4b123be57efab66f18e460e363e8f56e1a9c7.svg

note that it is welldefined, as 20230802-cokernel_of_a_group_802057973bafd24d0b9f406b3b509611a10c8b9f.svg with kernel as normal subgroup

Date: nil

Author: Anton Zakrewski

Created: 2024-10-13 So 15:35