Let's analyze the problem to determine the correct statements regarding cosets and subgroups.

Given Problem:

Let $H \subseteq G$ be a subgroup. The problem asks us to evaluate several statements about when a coset of $H$ (e.g., $gH$) forms a subgroup.

Key Points about Cosets and Subgroups:

1. Coset Definition: For $g \in G$, the left coset of $H$ in $G$ is $gH = \{ gh \mid h \in H \}$. Similarly, the right coset is $Hg = \{ hg \mid h \in H \}$.

2. Subgroup Criterion for Cosets:

   • A coset $gH$ forms a subgroup if and only if $gH = H$. This happens when $g \in H$, as $gH = H$ is equivalent to saying $g \in H$.

Analysis of Statements:

1. (A) "A coset is a subgroup if it contains $e$, in which case this coset is $H$."

   • True. If $gH$ contains the identity element $e$, then $gH = H$, which is a subgroup. The identity is always in $H$, so $g \in H$.

2. (B) "All cosets are subgroups."

   • False. Only the coset $H$ (i.e., when $g \in H$) is a subgroup. Other cosets are not subgroups in general.

3. (C) "The coset $gH$ is a subgroup if and only if $g = e$."

   • False. The correct condition is $g \in H$, not necessarily $g = e$.

4. (D) "The coset $gH$ is a subgroup if and only if $gH = H$."

   • True. This is the precise condition for a coset to be a subgroup.

5. (E) "The coset $gH$ is a subgroup if and only if $g \in H$."

   • True. If $g \in H$, then $gH = H$, which is a subgroup.

6. (F) "All cosets except for $H$ are not subgroups."

   • True. Only $H$ itself can be a subgroup.

7. (G) "The coset $eH = H$ is a subgroup."

   • True. By definition, $eH = H$, which is the subgroup $H$.

Correct Answers:

• (A), (D), (E), (F), and (G) are correct.

Let me know if you'd like further clarifications!

Related Questions:

1. What is the difference between a left coset and a right coset?
2. How can we prove that $gH$ is not a subgroup for $g \notin H$?
3. Can a coset $gH$ be a subgroup if $G$ is not a group?
4. Why is the identity element crucial in subgroup definitions?
5. How do left cosets and right cosets relate to normal subgroups?

Tip:

To verify subgroup properties, always check if the subset is closed under the group operation and inverses. For cosets, only $H$ itself satisfies these properties.