# finite group

• December 26th 2009, 07:43 AM
mms
finite group
Let G be a finite group, and let S and T be 2 subsets of G such that G does not equal ST. Show that $
\left| G \right| \geqslant \left| S \right| + \left| T \right|

$
• December 26th 2009, 12:55 PM
Opalg
Quote:

Originally Posted by mms
Let G be a finite group, and let S and T be 2 subsets of G such that G does not equal ST. Show that $
\left| G \right| \geqslant \left| S \right| + \left| T \right|

$

Take an element $g\notin ST$ and consider the set $S\cup gT^{-1}$.
• December 31st 2009, 08:33 AM
mms
Could you help me a little bit more? i still can't do this problem >.<

thanks!
• December 31st 2009, 08:58 AM
Opalg
There are |S| elements in S. There are |T| elements in $T^{-1}$, and also in its coset $gT^{-1}$. Also, the sets S and $gT^{-1}$ are disjoint: for suppose that an element $s\in S$ is also in $gT^{-1}$. Then $s = gt^{-1}$ for some $t\in T$. But that implies that $st=g$, which contradicts the choice of g. Therefore there are |S|+|T| distinct elements in the set $S\cup gT^{-1}$, and so $|G|\geqslant |S|+|T|$.
• December 31st 2009, 10:49 AM
mms
ahh i see, thank you!