I've only recently started looking at quotient groups, so I don't know if this question will make sense...

In this wiki article, $G/H$ is defined as the set of left cosets of $H$ in $G$, without any reference to whether or not $H$ is normal. In the quotient group article, however, the definition of is stated only for when $H$ is a normal subgroup of $G$.

  1. What accounts for the discrepancy between the two articles?
  2. Are quotient groups only defined when $H \triangleleft G$?
  3. If one is trying to determine what $G/H$ is explicitly, will this be affected if $H$ is normal or not? Are there any caveats I should anticipate if $H$ is not normal?

I do want to note that I came across a similar article, but I don't understand it all.

  • 7
    $\begingroup$ $G/H$ is a set in general. If $H$ is normal, it's also a group. $\endgroup$ – Qiaochu Yuan Jun 17 '13 at 20:36
  • 1
    $\begingroup$ You might enjoy this blog post by Gowers. $\endgroup$ – TTS Jun 17 '13 at 20:39
  • $\begingroup$ To add to Qiaochu's comment, $G/H$ is also always a $G$-space. $\endgroup$ – Rasmus Jun 17 '13 at 20:40
  • $\begingroup$ what part about the other post do you not understand? $\endgroup$ – Tobias Kildetoft Jun 17 '13 at 20:54

$G/H\,$ is quotient group if and only if $H$ is a normal subgroup of $G$. However, the notation $G/H$ denotes the set of left cosets of $H$ in $G$, and it does not necessarily denote a quotient group. If it also happens to be the case that $\;H \triangleleft G,\;$ then $G/H$ is not just the set of left cosets of $H$ in $G$, but also a group, namely, the quotient group, sometimes referred to as the factor group, the group of cosets under "coset multiplication", which of course is defined if and only if the left cosets of $H$ equal the right cosets of $H$ in $G$: i.e., if and only if $H$ is a normal subgroup of $G$.

  • $\begingroup$ I gave the OP a TU, you too! :-) $\endgroup$ – Amzoti Jun 18 '13 at 0:49
  • $\begingroup$ Love it+++++++++ $\endgroup$ – mrs Jun 18 '13 at 3:00
  • $\begingroup$ quotient group has puzzled me for along time when i saw it first time , it was hard to my mind to deal with sets as elements , it still puzzles me from time to time but less than the past. $\endgroup$ – Fawzy Hegab Jun 18 '13 at 3:23

G/H is the set of all left cosets of H in G but this is only a group if H is a normal subgroup. It is then a quotient group.

  • $\begingroup$ Oops you're right I got the slash wrong $\endgroup$ – user77904 Jun 17 '13 at 20:41

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