Normal Subgroups and Quotient Groups (aka Factor Groups) - Abstract Algebra

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Recommends AgentS · 2019-07 · Mathematics Stack Exchange · on “Criterion for cosets to act like a group”

I'm watching this [this video] video on quotient groups and came across the following(you don't have to click the link): $G$ is a group and $N$ is its subgroup. Using $N$ to make cosets : $N, g_1N, g_2N, g_3N, \ldots$ [LINK] I get why $x\in xN$ and $y\in yN$ , but I'm not able to figure out the last two lines . Any help understanding why they say $x\cdot y \in (xN)(yN)$ for cosets to act like a group? It is obvious right? (Since $x\in xN$ and $y\in yN$ , it is obvious $x\cdot y \in (xN)(yN)$ . I don't get how this makes the cosets a group...)

Recommends gen-ℤ ready to perish · 2020-05 · Mathematics Stack Exchange · on “How do I finish deriving this property of normal subgroups?”

Socratica has a fantastic video [this video] on normal subgroups and quotient groups, but there’s one part of which I can’t convince myself. Let $G$ be a commutative group under juxtaposition, let $N$ be a normal subgroup of $G$ , and let the quotient group $G/N$ be equipped with the operation $\cdot$ for clarity. The members of $G/N$ are of the form $gN$ , where $g\in G$ , and the quotient group operation is defined by $$xN\cdot yN = zN \iff \forall g_1\in xN\ \ \forall g_2\in yN\ :\ g_1g_2\in zN$$ We know that the coset containing $g_1g_2=g_1g_2e$ is $g_1g_2N$ since $e\in N$ . Therefore $\{g_1g_2N : g_1\in xN\wedge g_2\in yN\}\subseteq xN\cdot yN$ . However, the video claims $xyN = xN \cdot yN$

Recommends user856333 · 2021-10 · Mathematics Stack Exchange · on “Proving conditions for normal subgroups”

I was watching the following online lecture on normal subgroups, and came across the following proof: Normal and quotient groups [this video] If we have some group, G, and some subgroup of G, let's say N, we can construct left cosets like so: Each left coset: $gN$ for some $g \in G$ . Now we pick two cosets: $xN, yN$ . Now since $e \in N$ , $x.e = x \in xN$ and $y.e = y \in yN$ . Now here is the part that I don't understand: For cosets to act like a group: $x.y \in (xN)(yN)$ Why must this be true? And what is $(xN)(yN)$ .Is it the element-wise group operation of each element in $xN$ and $yN$ ? Then the proof said: $i.e. (xN)(yN) = xyN$ . This does not make sense to me, what is so obvious about th

Critical JohnIdlewood · 2019-05 · Mathematics Stack Exchange · on “Cosets form a group if normal subgroup”

Given a group G and its subgroup H that creates cosets, prove that cosets form a group iff H - is a normal subgroup. I've tried to find any good prove but I failed - most of the sources (including ProveWiki) provide this statement "as is" and other sources provide incorrect proves like here: [this video] 6:54 - brackets are opened as the operation is commutative - but in fact, it not necessarily true. Can you, please, tell why there's a huge leap between statement that product of two elements from two cosets must be contained by the third coset and this operation should be associative <=> subgroup ability to form up equal left and right cosets. Or give me, please, prove of the statement, I'll try