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Let G be a group and let H be a normal subgroup of G. Then prove that the rule of
coset multiplication

$(aH)(bH)$=$(ab)H$

gives a well defined binary operation on the set

$G/H=(aH| a \in G)$

Can anyone show me a way to approach this as I have no idea what this is asking

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  • $\begingroup$ The well-definedness comes into question when you ask: Suppose $a_i$ are different elements of $aH$, and $b_i$ different elements of $bH$. Is it really true that $(a_1H)(b_1H) = (a_2H)(b_2H)$? That is, is the calculation of $(aH)(bH)$ independent of the representatives chosen for each coset? Because there are (generally) lots of choices for representatives... $\endgroup$ – pjs36 Apr 21 '16 at 15:59
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    $\begingroup$ It might be instructive to work out by hand what happens in a simple case when $H$ isn't normal; the simplest such case is $G = S_3$ and $H = \langle (12) \rangle$. $\endgroup$ – Travis Willse Apr 21 '16 at 15:59
  • $\begingroup$ It is helpful to keep in mind that $H$ is normal in $G$ if and only if for every $g\in G$, we have $gHg^{-1} = H$, or equivalently $gH = Hg$. $\endgroup$ – Bungo Apr 21 '16 at 16:03
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The issue with working with quotient groups is that there are many representatives of the same coset. For example, in $\mathbb{Z}/5\mathbb{Z}$ one has that $$1+5\mathbb{Z}=\{\ldots,-9,-4,1,6,11,\ldots\}=6+5\mathbb{Z}$$ and $$2+5\mathbb{Z}=\{\ldots,-8,-3,2,7,12,\ldots\}=12+5\mathbb{Z}.$$ It is, of course, reasonable to be concerned whether $$3+5\mathbb{Z}=(1+5\mathbb{Z})+(2+5\mathbb{Z})=(6+5\mathbb{Z})+(12+5\mathbb{Z})=18+5\mathbb{Z}?$$ Of course, in this case everything works out just fine, but it is not always so. For example, take the subgroup $H=\langle(12)\rangle=\{(1),(12)\}\leq S_3$. We have $$(13)H=\{(13),(123)\}=(123)H$$ and $$(23)H=\{(23),(321)\}=(321)H$$ However, $(13)(23)H=(321)H$, while $(123)(321)H=(1)H=H$. Hence, $$(13)H(23)H=(13)(23)H=(321)H\neq H=(123)(321)H=(123)H(321)H$$ and the operation is not well defined.

The difference in the two cases is that $5\mathbb{Z}$ is a normal subgroup of $\mathbb{Z}$, while $H$ is not normal in $S_3$.

If we assume $H$ is a normal subgroup of $G$, we can show that the operation $aHbH=abH$ is well defined as follows:

Suppose $aH=cH$ and $bH=dH$. By definition, this means that $c^{-1}a\in H$ and $d^{-1}b\in H$. To show that $abH=cdH$, we need to show that $(cd)^{-1}(ab)\in H$.

Well, $$ (cd)^{-1}ab=d^{-1}c^{-1}ab=(d^{-1}(c^{-1}a)d)(d^{-1}b). $$ By assumption $d^{-1}b\in H$. Also, since $c^{-1}a\in H$ and $H$ is normal $d^{-1}(c^{-1}a)d\in H$. Finally, $H$ is a subgroup, so $(d^{-1}(c^{-1}a)d)(d^{-1}b)\in H$ and we're done.

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  • $\begingroup$ Correct me if I'm wrong, but what you showed is that aHbH=abH is well defined. But how do I show it is well defined on the set G/H? Don't i have to show that whatever I do, I always get an element in G/H $\endgroup$ – HueHue Apr 24 '16 at 9:22
  • $\begingroup$ That is automatic. $abH$ is an element of $G/H$ since $ab\in G$. $\endgroup$ – David Hill Apr 24 '16 at 15:29
  • $\begingroup$ So its enough just to prove that $cd^{-1}ab$ $\in H$ because a,b,ab $\in G$ $\endgroup$ – HueHue Apr 24 '16 at 16:02
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Since $H$ is normal, $Hb=bH$ and so: $$aHbH=a(Hb)H=a(bH)H=abHH=abH$$

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Given two cosets $aH, bH$, showing that the rule $(aH)(bH)=abH$ is well-defined amounts to showing that this product is independent of choice of coset representatives.

Let $a, a', b, b' \in G$ be such that $aH=a'H$ and $bH=b'H$.

We want to see that $(aH)(bH)=abH=a'b'H=(a'H)(b'H)$. It suffices to see $abH=a'b'H$.

Note first that $abH=ab'H$.

Since $aH=a'H$, there exists $h \in H$ such that $a=a'h$, so that $ab'H=a'hb'H$.

But $H$ is normal in $G$, so $(b')^{-1}hb'=h'$, for some $h' \in H$.

Then $hb'H=b'h'H=b'H$, so $abH=ab'H=a'hb'H=a'b'H$.

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You want to show that the coset H form a group $\frac{G}{H}$ under the binary operation $$(aH)(bH)=(ab)H$$ i.e it satisfy all the satisfies the following properties.

1.Closure.

2.associative law.

3.Identity law.

4.Inverse law.

Let H be a normal subgroup of G. Denote the set of all distinct cosets of H in G by $\frac{G}{H}$. Multiplication in $\frac{G}{H}$ is associative. Let $aH$ and $bH$ be arbitrary cosets of H in G.

$(aH)(bH)=a(bH)H=a(bH)H$ since H is normal.

$(aH)(bH)=(ab)HH=(ab)H^2$ Since $H^2=H$

$(aH)(bH)=(ab)H$

Thus $\frac{G}{H}$ is closed and $(aH)(bH)=(ab)H$

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  • $\begingroup$ This does not appear to answer the question. In particular, the main concern is that the operation is well-defined. $\endgroup$ – David Hill Apr 21 '16 at 16:16
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Since H is normal in G, abH=aHbH=a'Hb'H=a'b'H if aH=a'H and bH=b'H. Isn't it good enough?

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