# Isomorphism between factor group and cyclic group.

Is $$\mathbb{Z_4}\times\mathbb{Z_6}/\langle(0,1)\rangle$$ isomorphic to $$\mathbb{Z_6}$$? I have counted the order of the former is 4 whereas the order of latter is 6 . Is this the reason that I can conclude they are not isomorphic?

• Of course that's an excellent reason: they don't have the same cardinal, which is a rather trivial consequence (i.e., a necessary condition) of being isomorphic...whic is, among other things, a $\;1-1\;$ map ! – DonAntonio Jun 22 at 9:39
• So his two is isomorphic right ? Then how about $\mathbb{Z_6}\times\mathbb{Z_4}/\langle(0,1)\rangle$, are they isomorphic? – Ling Min Hao Jun 22 at 9:40
• Deleted my answer because @DonAntonio's is more than sufficient. – Ruben Jun 22 at 9:41
• See also this duplicate. – Dietrich Burde Jun 22 at 11:30

## 1 Answer

If $$\;A\cong B\;$$ , then there exists a bijective function $$\;f:A\to B\implies |A|=|B|\;$$ (cardinal = number of elements, if the sets are finite) are the same