Introduction:
In this question, we are given a group G and two subgroups H and K such that H has a finite index in G. We need to explain this concept in detail.

Definition:
The index of a subgroup H in a group G, denoted as [G : H], is the number of distinct cosets of H in G. It represents the number of different ways we can partition G into left cosets of H.

Explanation:
Here, we are given that H has a finite index in G. Let's break down the concept and explain it in detail:

1. Cosets:
A left coset of H in G is a set of the form gH, where g is an element of G. It contains all elements obtained by multiplying each element of H by g on the left. Similarly, a right coset of H in G is a set of the form Hg, where g is an element of G. We denote the left coset of H containing g as gH and the right coset of H containing g as Hg.

2. Partition of G:
The set of left cosets of H in G forms a partition of G. This means that every element of G belongs to exactly one left coset of H. Similarly, the set of right cosets of H in G forms a partition of G.

3. Index of H in G:
The index of H in G, denoted as [G : H], represents the number of distinct left cosets of H in G (or equivalently, the number of distinct right cosets of H in G). It is a measure of how "big" or "small" the subgroup H is in relation to the whole group G.

4. Finite Index:
If H has a finite index in G, it means that there are only a finite number of distinct left cosets of H in G (or equivalently, finite number of distinct right cosets of H in G). In other words, the number of ways we can partition G into left cosets of H is finite.

5. Applications:
The concept of finite index is useful in various areas of mathematics, including group theory, number theory, and algebraic geometry. It allows us to study the structure and properties of groups by considering their cosets and coset representatives. It also helps in proving important theorems, such as Lagrange's theorem which states that the order of a subgroup divides the order of the group.

Conclusion:
In this question, we have explained the concept of a subgroup H having finite index in a group G. The index of H represents the number of distinct cosets of H in G, and if this index is finite, it means that the number of ways we can partition G into cosets is finite. This concept has various applications in mathematics and helps in understanding the structure and properties of groups.