List questions from normal-subgroups

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To derive commutativeness of any group from the normality of all its subgroups & some other conditions

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Separability of conjugacy classes in conjugacy separable semidirect products.

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A subgroup such that at least one left coset is a right coset

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$H:=\{ g^2 : g \in G \}$ is a subgroup of $G$ $\implies $ $H$ is normal in $G$

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List Question

Proof: Every normal subgroup has corresponding Congruence relation and vice versa

How is this subgroup abelian?

Question on Subgroups and Left Cosets

A normal subgroup problem

To derive commutativeness of any group from the normality of all its subgroups & some other conditions

Separability of conjugacy classes in conjugacy separable semidirect products.

Does this property of normal subgroups have a name?

A subgroup such that at least one left coset is a right coset

Proving $H$ is a normal subgroup of $G$ if $H$ and $gH$ ($g\notin H$) are the only distinct left cosets

Define a normal subgroup of G

Normal subgroup of a group

$H \lhd N \lhd G$ but $H \ntriangleleft G$

Prove that a group is non-cyclic

Proving that the intersection of a Sylow p-group with a normal subgroup is also a Sylow p-group

$H:=\{ g^2 : g \in G \}$ is a subgroup of $G$ $\implies $ $H$ is normal in $G$

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