List questions from normal-subgroups

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Subgroup generated by subset of group

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Any Subgroup containing commutator subgroup is normal.

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A group with an infinite cyclic normal subgroup that has a finite cyclic quotient is abelian

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List Question

The normalizer of permutation

Index of subgroup of Real number in Complex Number

For element of order $2$, $\{1,z\}$ is normal subgroup iff $z \in Z(G)$

If $G = \langle A, N \rangle$ ... , show that $Z(G)$ has finite index.

Let $G\simeq N_1\times N_2$ while $N_i$ are normal subgroups of $G$. Then what can we say about those $N_i$?

Subgroup generated by subset of group

Any Subgroup containing commutator subgroup is normal.

Solutions of the equation $(m! + 2)\sigma(n) = 2n \cdot m!$ where $5 \leq m$

Property of prime divisors in group theory

Every element of a subgroup $N$ normalizes $N$

Prove that every subgroup of a free group is normal

Step in proof of equivalent definitions of normal subgroup

Show that $GL_n(\mathbb{F}_{23})$ has subgroup of index $2$

A group with an infinite cyclic normal subgroup that has a finite cyclic quotient is abelian

If $N/M$ is a normal subgroup of $G/M$ then $N$ is a normal in $G$

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