الجبر مساعدة لو ممكن

[لؤلؤة العلم]

السلام عليكم
لو ممكن عايزة حل لهذه الاسئلة:

prove that

every homophic image to an abelian group ia abelian but its converse is not true
********
every homophic image of acyclic group ia acyclic but its converse is not true
********
every quotient group of an abelian group ia abelian but its
converse is not true
********
every quotient group of acyclic group ia acyclic but its converse is not true
********​

والشكر الجزيل لكم​