Is Z2 is simple.

Question

Is Z2 simple.Does every subgroup of Z2 simple.

Solution.

Z2 is simple, however subgroup of order 1 in Z2 is not simple.

Answer 1

A group is simple when the only normal subgroups it has are the group itself and the trivial subgroup $\{e\}$. The group $\mathbb Z_2$ has only two subgroups altogether. So clearly every normal subgroup of $\mathbb Z_2$ is either $\mathbb Z_2$ or $\{e\}$. In other words, $\mathbb Z_2$ is simple. Its subgroup $\{e\}$ is also clearly simple.