# Subgroup index

• Nov 22nd 2009, 04:02 PM
bethh
Subgroup 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, 08:55 PM
Shanks
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, 10:20 PM
aliceinwonderland
Quote:

Originally Posted by bethh
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.

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

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

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

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

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