Normal subgroup having no common composition factor with its quotient group

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Definition

A subgroup $N$ of a group of finite composition length $G$ is termed a normal subgroup having no common composition factor with its quotient group if $N$ is a normal subgroup of $G$ and no composition factor of $N$ is isomorphic to any composition factor of the quotient group $G/N$.

Relation with other properties

Stronger properties

• Normal Sylow subgroup
• Normal Hall subgroup

Weaker properties

• Normal-homomorph-containing subgroup
• Characteristic subgroup
• Normal subgroup

Incomparable properties

• Complemented normal subgroup: For full proof, refer: No common composition factor with quotient group not implies complemented
• Normal subgroup having no nontrivial homomorphism to its quotient group: For full proof, refer: No common composition factor with quotient group not implies no proper nontrivial homomorphism to quotient group, No proper nontrivial homomorphism to quotient group not implies no common composition factor with quotient group
• Fully invariant subgroup