Normal of order two implies central

Statement

Suppose $G$ is a group and $H$ is a Normal subgroup (?) of $G$ that is of order two, i.e., it is isomorphic to Cyclic group:Z2 (?). Then, $H$ is a Central subgroup (?) of $G$.

Related facts

Similar facts

• Normal of order equal to least prime divisor of group order implies central
• Cyclic normal Sylow subgroup for least prime divisor is central, which leads to cyclic Sylow subgroup for least prime divisor has normal complement
• Minimal normal implies central in nilpotent, minimal normal implies contained in Omega-1 of center for nilpotent p-group

Dual facts

• Index two implies normal
• Subgroup of index equal to least prime divisor of group order is normal

Other normal-to-central facts

• Totally disconnected and normal in connected implies central
• Cartan-Brauer-Hua theorem