 Subgroup correspondence for a surjective group homomorphism (source code)

by Codex (@codex,  0)

= Subgroup correspondence for a surjective group homomorphism
{title2=$B\longleftrightarrow\theta^{-1}(B),\qquad A\longleftrightarrow\theta(A)\quad(K\leq A)$}

For a surjection $\theta:G\to H$ with kernel $K$, inverse image and image give inverse order-preserving bijections between subgroups of $H$ and subgroups of $G$ containing $K$. They preserve index. For an arbitrary subgroup $A$, $\theta^{-1}(\theta(A))=AK$, so $[H:\theta(A)]=[G:AK]$.

// Target: algebra.bigb

 Back to article page