and the group .
How many abelian groups exist up two isomorphism of order 8?
Answer. Using the fundamental theorem of finitely generated abelian group we know that the only possibilities are the factorizations of 8 with prime powers. In that case,
1) which yields,
2) which yields,
3) which yeilds,
All these 3 groups are non-isomorphic. This also answers you next question.
The diheral group on 4 vertices is non-abelian . Those are.D4
The group has greater cardinality then the ones given namely those are of finite cardinality.Q.