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I had this as a statement in my book, but I am unable to prove it using the four basic properties of a group:

  1. Closure
  2. Associativity
  3. Existence of identity
  4. Existence of inverses.

marked as duplicate by Dietrich Burde abstract-algebra Jun 15 at 10:59

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  • 3
    $\begingroup$ First you need 0. multiplication in $G/N$ is well-defined. $\endgroup$ – Lord Shark the Unknown Jun 15 at 9:06
  • 1
    $\begingroup$ Multiplication as in "group operation"?! @LordSharktheUnknown $\endgroup$ – Aakash Singh Bais Jun 15 at 9:11
  • $\begingroup$ Use the fact $G$ satisfies all these properties. $\endgroup$ – Sunny Jun 15 at 9:18
  • $\begingroup$ The only non trivial thing to check that this binary operation is well defined and rest of things are clear because elements of $G/N$ are representatives by the elements of $G$ which satisfies these properties. $\endgroup$ – Sunny Jun 15 at 9:30

Use that $G$ is group. For example, if $1 \in G$ denotes the identity element of $G$, we get

$(1N)(gN) = (1 \cdot g) N = gN = (g \cdot 1)N = (gN)(1N)$

for all $g \in G$. Thus $1N \in G/N$ is the identity element in the quotient group.


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