# proof that group homomorphisms preserve identity

###### Theorem.

A group homomorphism^{} preserves identity elements^{}.

###### Proof.

Let $\varphi :G\to K$ be a group homomorphism.
For clarity we use $\ast $ and $\star $ for the group operations^{} of $G$ and $K$, respectively. Also,
denote the identities^{} by ${1}_{G}$ and ${1}_{H}$ respectively.

By the definition of identity,

$${1}_{G}\ast {1}_{G}={1}_{G}.$$ | (1) |

Applying the homomorphism^{} $\varphi $ to (1) produces:

$$\varphi ({1}_{G})\star \varphi ({1}_{G})=\varphi ({1}_{G}\ast {1}_{G})=\varphi ({1}_{G}).$$ | (2) |

Multiply both sides of (2) by the inverse^{} of $\varphi ({1}_{G})$ in $K$,
and use the associativity of $\star $ to produce:

$$\varphi ({1}_{G})={(\varphi ({1}_{G}))}^{-1}\star \varphi ({1}_{G})\star \varphi ({1}_{G})={(\varphi ({1}_{G}))}^{-1}\star \varphi ({1}_{G})={1}_{K}.$$ | (3) |

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Title | proof that group homomorphisms preserve identity |
---|---|

Canonical name | ProofThatGroupHomomorphismsPreserveIdentity |

Date of creation | 2013-11-16 4:44:43 |

Last modified on | 2013-11-16 4:44:43 |

Owner | jacou (1000048) |

Last modified by | (0) |

Numerical id | 9 |

Author | jacou (0) |

Entry type | Proof |

Classification | msc 20A05 |

Synonym | 1234 |

Related topic | 1234 |

Defines | 1234 |