Let's proceed with the analysis step-by-step.

1. Determine the number of Sylow-7 subgroups of $G$.

Given the order of the group $|G| = 42$, we can factorize it as: $42 = 2 \times 3 \times 7$

The Sylow-7 subgroups of $G$ have orders equal to $7$.

According to the Sylow theorems, the number of Sylow-7 subgroups $n_7$ satisfies:

1. $n_7 \equiv 1 \mod 7$
2. $n_7$ divides $\frac{42}{7} = 6$

The possible divisors of 6 are 1, 2, 3, and 6. Among these, only 1 satisfies $n_7 \equiv 1 \mod 7$.

Therefore, there is exactly one Sylow-7 subgroup in $G$: $n_7 = 1$

2. Show that $G$ has a subgroup $H_2$ satisfying $|H_2| = 2$ and $H_1 \cap H_2 = \{e\}$.

Let $H_1 \subseteq G$ be a Sylow-3 subgroup of $G$, so $|H_1| = 3$.

We now consider the Sylow-2 subgroups of $G$. The number $n_2$ of Sylow-2 subgroups satisfies:

1. $n_2 \equiv 1 \mod 2$
2. $n_2$ divides $\frac{42}{2} = 21$

The possible values for $n_2$ are 1, 3, 7, and 21. Among these, 1 and 21 are the only values satisfying $n_2 \equiv 1 \mod 2$.

To show $H_1 \cap H_2 = \{e\}$:

• The order of $H_1$ is 3, and the order of any Sylow-2 subgroup $H_2$ is 2.
• Since 3 and 2 are coprime, their intersection $H_1 \cap H_2$ can only be the identity element $e$.

This implies there is at least one Sylow-2 subgroup $H_2$ of order 2 such that $H_1 \cap H_2 = \{e\}$.

3. Prove that $G$ has a subgroup of order 6.

A subgroup of order 6 can be constructed using the Sylow-3 and Sylow-2 subgroups.

Let $H_1 \subseteq G$ be a Sylow-3 subgroup (order 3) and $H_2 \subseteq G$ be a Sylow-2 subgroup (order 2).

Since $H_1 \cap H_2 = \{e\}$, the product $H_1H_2$ forms a subgroup. The order of $H_1H_2$ is: $|H_1H_2| = |H_1| \times |H_2| = 3 \times 2 = 6$

Thus, $G$ has a subgroup of order 6, which can be formed by combining the elements of the Sylow-3 and Sylow-2 subgroups.

Summary:

1. $G$ has exactly one Sylow-7 subgroup.
2. $G$ has a Sylow-2 subgroup $H_2$ such that $H_1 \cap H_2 = \{e\}$.
3. $G$ has a subgroup of order 6.

Do you want more details or have any questions?

Related Questions:

1. What are the Sylow theorems?
2. Can you give an example of a group of order 42?
3. How do Sylow subgroups help in classifying groups?
4. What is Lagrange's theorem in group theory?
5. How do you determine the order of a group element?
6. What are normal subgroups and how do they relate to Sylow subgroups?
7. Can a group have more than one Sylow subgroup of a given order?
8. How can the concept of cosets be used to understand group structure?

Tip: When dealing with Sylow subgroups, remember that their number has specific divisibility and congruence properties which are crucial for understanding the structure of the group.