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I am reading a paper by Kelly Ann Chambers titled " The isomophirms of the lattice of congruence relations on a group and the lattice of normal subgroups of a group".

Chambers is proving that in her Theorem that a mapping from the lattice of normal subgroups of a group G to the lattice of congruences on G is an isomorphism.

I understand the proof, however, I think that the author should also show that the image of the isomorphism $f$ is congruence everytime. However, I dont know how would I prove it. Do you have any idea how to proceed? Maybe show the properties of equivalence relations for the image of $f$ and than show that it is a subuniverse of $G$?

Any help is appreciated and here is the screenshot of the theorem.

Define $f\colon Nor(G)\to Con(G)$ by $f(N)=C_N$ where $aC_Nb$ if and only if $Na=Nb$ for all $a,b\in G$.

Theorem. The mapping $f\colon Nor(G)\to Con(G)$ given by $f(N)=C_N$ is a lattice isomorphism.

Arturo Magidin
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Tereza Tizkova
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    Not clear what you are asking. $C_N$ is an equivalence relation is true for any subgroup $N,$ not just normal subgroups. The paper probably skips this because it is a fact from intro group theory. – Thomas Andrews Nov 14 '21 at 19:24
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    Please would you type up the screenshot? Pictures of text aren't user friendly. – Shaun Nov 14 '21 at 19:25
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    Right, this is just the standard equivalence relation that defines cosets. – Randall Nov 14 '21 at 19:26
  • @ThomasAndrews I am asking how do we know that $C_N$ is congruence relation. This is true if $C_N$ is equivalence relation and subuniverse of $G$ I think, but I wasnt even aware why it is equivalence relation, how it can be proved. – Tereza Tizkova Nov 14 '21 at 19:29
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    It's a congruence relation because $N$ is normal subgroup: if $Na=Nb$ and $Nx=Ny$, then $Nax = NaNx = NbNy = Nby$. – Arturo Magidin Nov 14 '21 at 19:38
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    See also why do we define quotients for normal subgroups only for an extended discussion of the connection between congruences, equivalence relations, and normal subgroups in the context of groups. – Arturo Magidin Nov 14 '21 at 20:19
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    The paper states the following on line 18 of the first page: "Likewise, if $N$ is a normal subgroup of $G$, then the binary relation $C$, given by $aCb$ if and only if $Na=Nb$, for all $a,b\in G$, is a congruence relation on $G$." You really need to read the whole thing, not just pieces, if you want to understand it. If you were not sure of this assertion, then that's what you should ask, not whether the author is missing stuff. The author wasn't missing a proof: she stated, as a given in the introduction that this fact is clear. – Arturo Magidin Nov 15 '21 at 01:21

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