1
$\begingroup$

The theorem is stated as follows in the book:

Let $\phi:G\rightarrow G'$ be a group homomorphism, and let $H=Ker(\phi)$. Let $a\in G$. Then the set

$\phi^{-1}[\{\phi(a)\}] = \{x\in G | \phi(x)=\phi(a)\}$

is the left coset $aH$ of $H$, and is also the right coset $Ha$ of $H$. Consequently, the two partitions of $G$ into left cosets and into right cosets of $H$ are the same.

I'm trying to parse this statement and it's not clear to me what claim the author is trying to make at the very end when he says "Consequently, the two partitions of $G$ into left cosets and into right cosets of $H$ are the same." I'm under the impression that, in general, the left and right cosets are not always the same. Under what condition are they the same? Under the condition that you have a homomorphism?

Let me mention that at this point, we're not supposed to know what a normal subgroup is. The author introduces the idea of a normal subgroup 2 pages later.

  • 1
    $\forall a\in G,aH=Ha$ is precisely the condition of being a **normal** subgroup: this shows kernels of homomorphisms are always normal subgroups. (In fact, the converse is true: every normal subgroup $H\trianglelefteq G$ is the kernel of some homomorphism, in particular the quotient map $G\to G/H$.) So you are right, left and right cosets are not generally the same. It should be clear what it means for two partitions of a set $S$ to be the same: each partition is a set of disjoint subsets of $S$ whose union is $S$, and two partitions are the same when they are the same set of subsets.2012-10-02
  • 0
    In response to your edit: you may not need to know what the definition of a normal subgroup is in order to do your homework or read your text, but normality *is the answer to your question* "under what condition are they the same?" (Although to nitpick, it is technically a potentially different situation for each left coset $aH$ to be *the* right coset $Ha$ versus each left coset $aH$ be *some* right coset $Hb$. I believe they have the same answer regardless, but the latter is more work.)2012-10-02
  • 1
    If, because of normality, you know that $aH$ is also a right coset, then since $Ha$ is the only right coset that contains $a$, it follows as the night the day that $aH=Ha$.2012-10-02

3 Answers 3