I'm learning algebra by myself and this concept is confusing me. Please excuse me if I define anything wrong... I've never expressed myself in this language before.(adsbygoogle = window.adsbygoogle || []).push({});

Lets say we have a group [itex]G[/itex] and a group [itex]G'[/itex] and there exists a homomorphism [itex]R: G → G'[/itex] and for any element [itex]g \in G[/itex], the equivalence class of g is denoted as [itex][g]_{R} = \{h \in G \:|\: f(h) = f(g)\}[/itex]

I understand the factor space [itex]G/R[/itex] as the set of all equivalence classes of [itex]G[/itex]:

[itex]G/R = \{[g]_{R} \:|\: g \in G\}[/itex]

but another way I always see this explained (that I'm not clear on) is if we have a subgroup [itex]H \subset G[/itex] then we can define a factor space with left cosets.

[itex]G/H = \{gH \:|\: g \in G\}[/itex]

How are these definitions stating the same thing? Does it have something to do with [itex]H[/itex] being the kernel of a homomorphism? I don't really understand what cosets have to do with equivalence relations.

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# Factor (Quotient) Space definitions.

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