Group Extension

Definition

Let $G,K:\mathrm{Ab}$ be an Abelian group. A group extension of $G$ by $K$ is a group $E$ that fits into a short exact sequence of the form:

$1\to K\stackrel{i}{\to }E\stackrel{p}{\to }G\to 1$

where $i$ is an injective homomorphism, $p$ is a surjective homomorphism, and $\text{im}(i)=\text{ker}(p)$. In this context, $K$ is isomorphic to a normal subgroup of $E$, and the quotient $E/K$ is isomorphic to $G$.

Equivalence of Extensions

Two extensions $E$ and $E^{\prime }$ of $G$ by $K$ are equivalent if there exists a group isomorphism $\varphi :E\to E^{\prime }$ such that the following diagram commutes:

1 \to K \to &E \to G \to 1 \\ \downarrow \text{id}_K \quad &\downarrow \phi \quad \downarrow \text{id}_G \\ 1 \to K \to &E' \to G \to 1 \end{aligned}$$ ### Classification Extensions are classified by the second [[cohomology group]] $H^2(G, K)$ when $K$ is Abelian. If the sequence [[Split Morphism (Category Theory)|splits]], meaning there exists a [[Section (Category Theory)|section]] $s: G \to E$ such that $p \circ s = \text{id}_G$, the extension is a *[[semidirect product]]* $K \rtimes_\theta G$. If the extension is central, $K$ lies in the center of $E$, denoted $Z(E)$. ## References 1. Brown, K. S. (1982). *Cohomology of Groups*. Springer-Verlag. 2. Rotman, J. J. (1995). *An Introduction to the Theory of Groups*. Graduate Texts in Mathematics.