Suppose H and K are distinct subgroups of G of index 2. Prove that H intersect K is a normal subgroup of index 4 and that G/(H intersect K) is not cyclic.

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- Nov 22nd 2009, 03:02 PMbethhSubgroup index
Suppose H and K are distinct subgroups of G of index 2. Prove that H intersect K is a normal subgroup of index 4 and that G/(H intersect K) is not cyclic.

- Nov 22nd 2009, 07:55 PMShanks
To prove H n K is a normal subgroup of index 4, you need to prove HK=G;

and if it is cyclic, then H=K. - Nov 22nd 2009, 09:20 PMaliceinwonderland
$\displaystyle \frac{H}{H \cap K} \triangleleft \frac{G}{H \cap K}$ since its index is 2.

$\displaystyle \frac{K}{H \cap K} \triangleleft \frac{G}{H \cap K}$ since its index is 2.

$\displaystyle \frac{H}{H \cap K} \cap \frac{K}{H \cap K}=1$.

By second isomorphism theorem, $\displaystyle \frac{H}{H \cap K} \cong \frac{HK}{K} \cong \frac{G}{K} \cong \mathbb{Z}_2$. Similary, $\displaystyle \frac{K}{H \cap K} \cong \mathbb{Z}_2$.

It follows that $\displaystyle \frac{G}{H \cap K} \cong \frac{H}{H \cap K} \oplus \frac{K}{H \cap K} \cong \mathbb{Z}_2 \oplus \mathbb{Z}_2 $, which is isomorphic to Klein-4 group. Thus $\displaystyle \frac{G}{H \cap K}$ is not cyclic.