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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.

Community
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TheRealFakeNews
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2 Answers2

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$G/H\,$ 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 $G$.

amWhy
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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.

user77904
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