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. De Beer Refinish Icris Software.epubl.Q:
Show that kernel of a homomorphism is a subgroup
Let $G$ be a group and $G \leq H$ a homomorphism, how to show that $ker(G)$ is a subgroup of $G$?
A:
A subgroup $K$ of $G$ must contain the identity $1$ of $G$. Since a homomorphism respects the identity, $1\in H$, $H$ contains $1$. In particular, if $1\in G\le H$, then $1\in H$, so that $1\in K$. This proves that $K$ is a subgroup of $G$.
Gmina Zambrów, Łódź Voivodeship
Gmina Zambrów is a rural gmina (administrative district) in Łęczyca County, Łódź Voivodeship, in central Poland. Its seat is the town of Zambrów, although the town is not part of the territory of the gmina.
The gmina covers an area of, and as of 2006 its total population is 13,908.
The gmina contains part of the protected area called Łódź Lake District Landscape Park.
Villages
Gmina Zambrów contains the villages and settlements of Antonieczki, Barłowice, Barwice, Belina, Bezdąznica, Bezdąznica-Kolonia, Bogowa Jasna, Bosia, Brześć, Bruszkowo, Dębina, Godlewo, Kamińska Huta, Kamińska Huta-Lesie, Kazimierzew, Kazimierzew-Kolonia, Kępa Różańska, Kępa Różańska-Kolonia, Kolonia, Kukierz, Kukierz-Żukowice, Nasiądz, Obcina, Okolice, Osogólnia, Pańska Huta, Parkowa Huta, Pisk
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