Quotient Group by Intersection of Normal Subgroups not necessarily Cyclic if Quotient Groups are
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Theorem
Let $G$ be a group.
Let $N$ and $K$ be normal subgroups of $G$.
Let the quotient groups $G / N$ and $G / K$ be cyclic.
Then the quotient group $G / \paren {N \cap K}$ is not necessarily cyclic.
Proof
Let $G = \set {e, a, b, c}$ be the Klein $4$-group whose identity element is $e$.
Let $N = \set {e, a}$ and $K = \set {e, b}$.
By Subgroups of Klein Four-Group, both $N$ and $K$ are subgroups of $G$.
By Prime Group is Cyclic, both $N$ and $K$ are cyclic.
By Subgroup of Abelian Group is Normal, both $N$ and $K$ are normal in $G$.
Then we have that:
- $N \cap K = e$
and so by Trivial Quotient Group is Quotient Group:
- $G / \paren {N \cap K} \cong G$
But $G$ is the Klein $4$-group, which is not cyclic.
$\blacksquare$
Sources
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $8$: Homomorphisms, Normal Subgroups and Quotient Groups: Exercise $16$