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# Normal Subgroup: Definition, Examples Abelian, Cyclic Subgroups

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Let $G$ be a group with neutral​ element $e$.

A normal subgroup $N$ of $G$, written $N\vartriangleleft G$, is a subgroup such that $gN=Ng$, for all $g\in G$.

The center of a group is the set of elements of $G$ that commute with every element of $G$.

A cyclic group is a group consisting of all powers $g^n$, $n\in\mathbb{Z}$, of a single element $g$.

Which of the following are false in general?

Select ALL that apply.

A

Every subgroup of an abelian group is normal.

B

Every abelian subgroup of a group is normal.

C

The center of a group is normal.

D

Every nontrivial group has at least two normal subgroups.

E

Every cyclic subgroup of a group is normal.