Misha Gavrilovich’s paper Expressing the statement of the Feit-Thompson theorem with diagrams in the category of finite groups, available from
- http://arxiv.org/abs/1608.05927, or
is a follow-up to his paper in The De Morgan Gazette,
M. Gavrilovich, Point-set topology as diagram chasing computations, The De Morgan Gazette, 5 no. 4 (2014), 23-32
The paper raises important questions about optimal approaches to exposition of elementary group theory: quite a number of group-theoretic concepts (for example, solvable, nilpotent group, p-group and prime-to-p group, abelian, perfect, subnormal subgroup, injective and surjective homomorphism) can be expressed as diagram chasing in the category theoretic language.