If Cylic subgroup implies abelian implies normal then how A5 is simple group [closed]

I am facing problem $A_5$ is simple group, but $A_5$ had 10 cyclic subgroup of order 3, from cyclic $\Rightarrow$ abelian $\Rightarrow$ normal we can say $A_5$ has 10 normal subgroups, but $A_5$ is normal

Help me with this doubt

Solution 1:

A cyclic group is abelian, and in an abelian group, every subgroup is normal.

However, the abelianness of the subgroup itself says relatively little about its normality. A subgroup being abelian only says something about how the elements of the subgroup interact with one another. A subgroup being normal says something about how the elements of the subgroup interact with the rest of the elements of the (super)group.

There are plenty of abelian subgroups that aren't normal. For instance, the subgroup generated by a 2-cycle in $S_3$. Or, as you found, the 10 subgroups generated by 3-cycles in $A_5$.