ShortExactSequence

The information you've provided forms a solid foundation for understanding key concepts in group theory, including exact sequences, centralizers, centers, and group products. Here are further details:

1. Exact Sequences: As mentioned, a sequence of groups ${G_0 = 1, G_1, ..., G_k, G_{k+1} = 1}$ with morphisms $\theta_j: G_j \rightarrow G_{j+1}$ for $0 \leq j \leq k$ forms an exact sequence if the image of each morphism $\theta_{j−1}$ (which is a subgroup of $G_j$) is the kernel of the next morphism $\theta_j$. This concept expresses a certain balance or compatibility between the groups and the morphisms in the sequence.

   A short exact sequence is a special kind of exact sequence where $k=3$. In such a sequence, the first morphism is injective, the last morphism is surjective, and the image of any morphism is the kernel of the next. This captures many interesting properties about the groups and morphisms involved.

2. Centralizers and Centers: The centralizer of a subset $A$ of a group $G$, denoted by $C_G(A)$, is the set of all elements in $G$ that commute with every element in $A$. The center of a group $G$, denoted by $Z(G)$, is the set of elements that commute with every element in $G$. It's a normal subgroup of $G$.

3. Product of Groups: If $G_1$ and $G_2$ are groups, then their product $G_1 \times G_2$ is a group whose operation is defined componentwise. This means that the product of two pairs $(g_1, g_2)$ and $(h_1, h_2)$ in $G_1 \times G_2$ is $(g_1h_1, g_2h_2)$. The subgroups $G_1 \times {e_2}$ and ${e_1} \times G_2$ are isomorphic to $G_1$ and $G_2$, respectively, which explains how $G_1$ and $G_2$ sit inside their product.

This information provides some important tools for studying the structure of groups and the relationships between different groups.