I will solve the problems one by one in detail. Let's begin with the first exercise.

1. Problem:

Let $H \leq G$ (subgroup). Show that $K = \bigcap_{g \in G} gHg^{-1}$ is a normal subgroup of $G$.

Solution:

1. $K$ is defined as the intersection of all conjugates of $H$ in $G$: $K = \bigcap_{g \in G} gHg^{-1}.$ This means $K \subseteq gHg^{-1}$ for all $g \in G$.

2. Closure of $K$:
   Since $K$ is a subgroup of $G$, it is closed under group operations (multiplication and taking inverses).

3. Normality of $K$:
   To prove $K \trianglelefteq G$, we need to show that $gKg^{-1} = K$ for all $g \in G$.

   • Take $x \in K$. Then $x \in g'Hg'^{-1}$ for all $g' \in G$.
     Conjugating $x$ by $g$, we have: $gxg^{-1} \in g(g'Hg'^{-1})g^{-1}.$ Using associativity of group operations: $g(g'Hg'^{-1})g^{-1} = (gg')H(gg')^{-1}.$ Hence, $gxg^{-1} \in \bigcap_{g' \in G} g'Hg'^{-1} = K$.
     This proves $gKg^{-1} \subseteq K$. Similarly, $K \subseteq gKg^{-1}$, so $gKg^{-1} = K$.

Thus, $K$ is a normal subgroup of $G$.

2. Problem:

Let $H \leq G$. If $g^2 \in H$ and $gHg^{-1} = H$, then show $G/H$ is abelian.

Solution:

1. $G/H$ is the set of cosets of $H$ in $G$, and we want to show that the operation on these cosets is commutative.

2. Let $g, h \in G$, and consider the cosets $gH$ and $hH$.
   Their product in $G/H$ is: $(gH)(hH) = ghH.$

3. To prove commutativity, we need to show: $ghH = hgH.$

4. Given that $gHg^{-1} = H$, it follows that $gH = Hg$, and similarly $hH = Hh$.

5. Also, since $g^2 \in H$, we have $g \cdot g = g^2 \in H$, so $g$ behaves like an element of $H$ under coset multiplication.

6. Since all elements commute within $H$, and $gHg^{-1} = H$, this implies $gh = hg$ mod $H$.

Thus, $G/H$ is abelian.

Would you like to proceed with the next problems or have questions about these?

Follow-up Questions:

1. What does $gHg^{-1}$ represent in a group structure?
2. Can we generalize the result of Problem 1 to infinite groups?
3. How does $G/H$ being abelian affect the group $G$?
4. What is the significance of normal subgroups in group theory?
5. Are there examples of $G/H$ not being abelian when conditions differ?

Tip: Normal subgroups are critical when studying the quotient structure of groups—ensure you understand their properties fully!