# proof of second isomorphism theorem for groups

First, we shall prove that $HK$ is a subgroup^{} of $G$:
Since $e\in H$ and $e\in K$, clearly $e={e}^{2}\in HK$.
Take ${h}_{1},{h}_{2}\in H,{k}_{1},{k}_{2}\in K$.
Clearly ${h}_{1}{k}_{1},{h}_{2}{k}_{2}\in HK$.
Further,

$${h}_{1}{k}_{1}{h}_{2}{k}_{2}={h}_{1}({h}_{2}{h}_{2}^{-1}){k}_{1}{h}_{2}{k}_{2}={h}_{1}{h}_{2}({h}_{2}^{-1}{k}_{1}{h}_{2}){k}_{2}$$ |

Since $K$ is a normal subgroup^{} of $G$ and ${h}_{2}\in G$,
then ${h}_{2}^{-1}{k}_{1}{h}_{2}\in K$.
Therefore ${h}_{1}{h}_{2}({h}_{2}^{-1}{k}_{1}{h}_{2}){k}_{2}\in HK$,
so $HK$ is closed under multiplication^{}.

Also, ${(hk)}^{-1}\in HK$ for $h\in H$, $k\in K$, since

$${(hk)}^{-1}={k}^{-1}{h}^{-1}={h}^{-1}h{k}^{-1}{h}^{-1}$$ |

and $h{k}^{-1}{h}^{-1}\in K$ since $K$ is a normal subgroup of $G$.
So $HK$ is closed under inverses^{}, and is thus a subgroup of $G$.

Since $HK$ is a subgroup of $G$, the normality of $K$ in $HK$ follows immediately from the normality of $K$ in $G$.

Clearly $H\cap K$ is a subgroup of $G$,
since it is the intersection^{} of two subgroups of $G$.

Finally, define $\varphi :H\to HK/K$ by $\varphi (h)=hK$.
We claim that $\varphi $ is a surjective homomorphism^{} from $H$ to $HK/K$.
Let ${h}_{0}{k}_{0}K$ be some element of $HK/K$;
since ${k}_{0}\in K$, then ${h}_{0}{k}_{0}K={h}_{0}K$, and $\varphi ({h}_{0})={h}_{0}K$.
Now

$$\mathrm{ker}(\varphi )=\{h\in H\mid \varphi (h)=K\}=\{h\in H\mid hK=K\}$$ |

and if $hK=K$, then we must have $h\in K$. So

$$\mathrm{ker}(\varphi )=\{h\in H\mid h\in K\}=H\cap K$$ |

Thus, since $\varphi (H)=HK/K$ and $\mathrm{ker}\varphi =H\cap K$,
by the First Isomorphism Theorem^{} we see that
$H\cap K$ is normal in $H$
and that there is a canonical isomorphism between $H/(H\cap K)$ and $HK/K$.

Title | proof of second isomorphism theorem for groups |
---|---|

Canonical name | ProofOfSecondIsomorphismTheoremForGroups |

Date of creation | 2013-03-22 12:49:47 |

Last modified on | 2013-03-22 12:49:47 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 17 |

Author | yark (2760) |

Entry type | Proof |

Classification | msc 20A05 |

Related topic | ProofOfSecondIsomorphismTheoremForRings |