List questions from normal-subgroups

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List Question

Group Z/3Z Proof of Subgroup.

Showing normality with just generators

Ambiguity of the notation $\lhd$

The normal subgroup of G generated by the set A

Can a group have two different subgroups of index $2$?

Show that $\phi : H\to HN/N$ defined by $\phi(h)=hN$ is injective.

Normal Fuchsian subgroup of $PSL(2,\mathbb{R})$

Every finite group has a a sequence of nested subgroups where adjacent elements are normal

What does normality is image closed mean here?

Normal closure of a subgroup that is identical to its commutator subgroup inherits this property

Does $P^g=\{gpg^{-1}|g \in G\}$ have the same order as P, where |P|=7

Even and odd elements in more general additive groups

Show that $\phi$ induces a canonical homomorphism.

Defining a normal subgroup to be maximal.

Show $\phi:H\to HN/N$ is one-to-one.

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