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Abstract Algebra

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Isomorphism Theorems: Subgroup Property from Order & Intersection

ABSALG-VX1KRY

Let $G$ be a group and let $H$ and $K$ be subgroups of $G$.

Suppose the orders $|G|$, $|H|$, and $|K|$ of these groups satisfy:

$15$ divides $|G|$ and $\cfrac1{15}|G|\le 15$, $\qquad|H|=4$, $\quad|K|=8$, $\quad |H\cap K|=2$

Which of the following are true?

Select ALL that apply.​

A

$H$ is a normal subgroup of $G$.

B

$K$ is the kernel of a group homomorphism of $G$.

C

Neither $H$ nor $K$ are normal subgroups of $G$.

D

$H/(H\cap K)$ is the image of a group homomorphism with kernel isomorphic to $K$.