# Tower Law for Subgroups/Proof 1

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## Theorem

Let $\struct {G, \circ}$ be a group.

Let $H$ be a subgroup of $G$ with finite index.

Let $K$ be a subgroup of $H$.

Then:

- $\index G K = \index G H \index H K$

where $\index G H$ denotes the index of $H$ in $G$.

## Proof

Let $p = \index G H$, $q = \index H K$.

By hypothesis these numbers are finite.

Therefore, there exist $g_1, \ldots, g_p \in G$ such that $G$ is a disjoint union: $\ds G = \bigsqcup_{i \mathop = 1}^p g_i H$

Similarly, there exist $h_1,\ldots,h_q \in H$ such that $H$ is a disjoint union: $\ds H = \bigsqcup_{j \mathop = 1}^q h_j K$

Thus:

\(\ds G\) | \(=\) | \(\ds \bigsqcup_{i \mathop = 1}^p g_i H\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \bigsqcup_{i \mathop = 1}^p g_i \bigsqcup_{j \mathop = 1}^q h_j K\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \bigsqcup_{i \mathop = 1}^p \bigsqcup_{j \mathop = 1}^q g_i \paren {h_j K}\) | Product of Subset with Union | |||||||||||

\(\ds \) | \(=\) | \(\ds \bigsqcup_{i \mathop = 1}^p \bigsqcup_{j \mathop = 1}^q \paren {g_i h_j} K\) | Subset Product within Semigroup is Associative: Corollary |

This expression for $G$ is the disjoint union of $p q$ cosets.

Therefore the number of elements of the coset space is:

- $\index G K = p q = \index G H \index H K$

$\blacksquare$

## Sources

- 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $2$: Subgroups and Cosets: $\S 39$