Second Isomorphism Theorem (Groups)

For H ≤ G and K ⊲ G, there is a natural isomorphism H/(H∩K) ≅ HK/K

Second Isomorphism Theorem (Groups)

Second Isomorphism Theorem (Groups). Let $G$ be a group , let $H \le G$ be a subgroup , and let $K \trianglelefteq G$ be a normal subgroup . Define the subset

HK = \{hk : h \in H,\ k \in K\}.

Then $HK \le G$ , $K \trianglelefteq HK$ , and $H \cap K \trianglelefteq H$ . Moreover, the map

\phi: H \to HK/K, \qquad \phi(h) = hK,

is a homomorphism with kernel $H \cap K$ , hence induces an isomorphism of quotient groups

H/(H \cap K) \cong HK/K.

This theorem compares a subgroup with its image in a quotient and is frequently used to compute or identify quotients inside a larger group. It is most efficiently proved by applying the first isomorphism theorem to the restriction of the quotient map $HK \to HK/K$ .

Proof sketch. The product set $HK$ is a subgroup because $K$ is normal, so products and inverses stay in $HK$ . The quotient map $HK \to HK/K$ restricts to $H \to HK/K$ with kernel exactly $H \cap K$ . Apply the first isomorphism theorem to obtain $H/(H\cap K) \cong HK/K$ .