Currently in Abstract Algebra, discussing group theory. In order to show two groups are isomorphic to each other, I know what you need to show, $1$-$1$, onto, and homomorphism. what I'm having a difficult time doing however is creating a map between the two groups.
Could anyone make any suggestions?
Specifically showing:
$$(Z_2 \times Z_4)/\langle(0,2)\rangle\cong Z_4$$
Since those two groups are not isomorphic, you will have trouble finding an isomorphism. For any element $g\in(\mathbb Z_2\times\mathbb Z_4)/\langle(0,2)\rangle$, you have $g+g=0$, while this is not true for $\mathbb Z_4$.
The question of abelian groups turns out to be entirely determined by the number of elements of order $d$ for each $d$.
The general question of determining if two groups are isomorphic (not even finding the isomorphism) is of unknown complexity. As of $2011$, the question has upper bound of $n^{\log n + O(1)}$ time.