Quotient Group of Direct Products

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $G$ and $G'$ be groups.

Let:

$H \lhd G$
$H' \lhd G'$

where $\lhd$ denotes the relation of being a normal subgroup.


Then:

$\paren {G \times G'} / \paren {H \times H'}$ is isomorphic to $\paren {G / H} \times \paren {G' / H'}$

where:

$G \times G'$ denotes the group direct product of $H$ and $H'$
$G / H$ denotes the quotient group of $G$ by $H$.


Proof

Let $\phi_1: G \to G / H$ and $\phi_2: G' \to G' / H'$ be the quotient epimorphisms with $H$ and $H'$ as their kernels, respectively.

Now define a homomorphism $\phi: G \times G' \to \paren {G / H} \times \paren {G' / H'}$ by:

$\phi = \phi_1 \times \phi_2$

so:

$\map \phi {\tuple {x, x'} } = \tuple {\map {\phi_1} x, \map {\phi_1} {x'} }$

The kernel of $\phi$ is clearly $H \times H'$, and $\phi$ is surjective.

So by the First Isomorphism Theorem for Groups:

$\paren {G / H} \times \paren {G' / H'} \cong \paren {G \times G'} / \paren {H \times H'}$

$\blacksquare$


Sources