another group theory question

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Nov 14th 2010, 05:03 AM
situation

another group theory question

Let G be a group and H be a subgroup of G. For any $g_1 , g_2 \in G$ prove that:
$g_2 \in g_1H \Rightarrow g_1H = g_2H$
Use this to prove that any two left cosets of H are equal or disjoint.

help would be much appreciated! thanks :)
Nov 14th 2010, 07:48 AM
Drexel28

Quote:

Originally Posted by situation

Let G be a group and H be a subgroup of G. For any $g_1 , g_2 \in G$ prove that:
$g_2 \in g_1H \Rightarrow g_1H = g_2H$
Use this to prove that any two left cosets of H are equal or disjoint.

help would be much appreciated! thanks :)

I'm sure you're aware that $(ab)H=a(bH)$ and $hH=H$ for all $h\in H$ . So, note that $g_2\in g_1 h\implies g_2=g_1h$ for some $h\in H$ and so $g_1h^{-1}=g_2$ . Thus, $g_2H=(g_1h^{-1})H=g_1(h^{-1}H)=g_1H$
Nov 15th 2010, 03:36 AM
situation

Quote:

Originally Posted by Drexel28

I'm sure you're aware that $(ab)H=a(bH)$ and $hH=H$ for all $h\in H$ . So, note that $g_2\in g_1 h\implies g_2=g_1h$ for some $h\in H$ and so $g_1h^{-1}=g_2$ . Thus, $g_2H=(g_1h^{-1})H=g_1(h^{-1}H)=g_1H$

thanks man, just a quick one about the result $hH=H$ how is this the case? if you wouldnt mind explaining that further it would be much appreciated! :)
Nov 15th 2010, 03:49 AM
Drexel28

Quote:

Originally Posted by situation

thanks man, just a quick one about the result $hH=H$ how is this the case? if you wouldnt mind explaining that further it would be much appreciated! :)

No problem! By definition $hH=\left\{hh':h'\in H\right\}$ clearly then since each element is the multiplication of elements in $H$ we have that $hH\subseteq H$ . But, if $h'\in H$ then $h^{-1}h'\in H$ and so $h(h^{-1}h')=h'\in hH$ and so $H\subseteq hH$ . Therefore, $H=hH$ (this is a result usually proven along with Cayley's theorem, that the map $\sigma_h:H\to H:h'\mapsto hh'$ is a permutation)