Question

Let \(H\) be a subgroup of a finite group \(G\) and \(K\) a subgroup of \(H\). Suppose that the index \([G: H]=n\) and the index \([H: K]=m .\) Show that the index \([G: K]=n m\). (Hint: Let \(x_{i} H\) be the distinct left cosets of \(H\) in \(G\) and \(y_{j} K\) the distinct left cosets of \(K\) in \(H\). Show that \(x_{i} y_{j} K\) are the distinct left cosets of \(K\) in \(G\).)

Short answer

The index \([G: K]\) is \(n \times m\).

Step-by-step solution

Identifying Cosets in G

Given that the index \([G:H]=n\), identify that \(H\) has \(n\) distinct left cosets in \(G\), denoted as \(x_i H\) for \(i = 1, 2, \, \ldots, n\). These are the distinct sets formed when multiplying \(G\)'s elements by \(H\).

Identifying Cosets in H

Since \([H:K]=m\), \(K\) has \(m\) distinct left cosets in \(H\), represented by \(y_j K\) for \(j = 1, 2, \, \ldots, m\). Each element of \(H\) is accounted for in one of these cosets.

Constructing Cosets in G Relative to K

Observe that elements of \(G\) can be rewritten as \(x_i y_j K\), where \(x_i\) spans the cosets of \(H\) in \(G\), and \(y_j\) spans the cosets of \(K\) in \(H\). This forms new cosets \(x_i y_j K\).

Counting Total Cosets in G

Since we have established the cosets \(x_i y_j K\), notice there are \(n\) choices for \(x_i\) and \(m\) choices for \(y_j\), resulting in \(n \times m\) distinct combinations. Thus, \([G:K] = nm\) follows as we have accounted for all elements in distinct cosets.

Key concepts

Subgroup

A subgroup is a smaller group contained within a larger group, possessing the same operation and obeying all group laws as the larger group. To be a subgroup of a group \( G \), a set \( H \) must:

• Include the identity element of \( G \).
• Be closed under the group operation (if \( h_1, h_2 \in H \), then \( h_1 \, h_2 \in H \)).
• Have inverses for every element within it (if \( h \in H \), then the inverse \( h^{-1} \in H \)).

Understanding subgroups allows mathematicians to break a problem down into a smaller, more manageable structure. For example, if \( K \) is a subgroup of another group, \( H \), which is itself a subgroup of a larger group, \( G \), then \( K \) retains the properties and structure conducive to \( H \) and \( G \). This hierarchy assists in analyzing and studying intrinsic group properties.

Finite Group

Finite groups are those with a finite number of elements. The number of elements in a group is its order. This concept is crucial in Group Theory because it influences the structure and study of a group. For any finite group \( G \), one can observe features like:

• Simple group analysis, where breaking down groups into smaller components is feasible.
• Efficient computations of group elements and operations.
• Pivot to applications such as symmetry operations in molecules, or group actions in permutations.

In the context of the exercise, both \( G \) and its subgroups \( H \) and \( K \) are finite. Recognizing the finitude of these groups ensures straightforward calculations of cosets and indices, ultimately enabling us to easily establish relationships like \([G:K] = [G:H] \, [H:K] \).

Cosets

Cosets are essential in understanding the partitioning of a group concerning a subgroup. If \( H \) is a subgroup of \( G \), the left coset of an element \( g \in G \) relative to \( H \) is the set \( gH = \{gh : h \in H \} \). Cosets have some remarkable properties:

• They partition the group \( G \) into disjoint sets.
• Every element of \( G \) belongs to exactly one coset of \( H \).
• The size of each coset is equal to the order of \( H \).

In our exercise, we find that \( H \) has \( n \) distinct left cosets in \( G \) (i.e., \( x_i H \)), while \( K \) has \( m \) cosets in \( H \) (i.e., \( y_j K \)). By multiplying these cosets, we can build the cosets of \( K \) in \( G \), demonstrating that the connection between indices gives us a comprehensive understanding of group structure.

Index of a Subgroup

The index of a subgroup provides insight into how many times a subgroup fits into its parent group. Given a group \( G \) and its subgroup \( H \), the index \([G:H]\) is defined as the number of distinct left cosets of \( H \) in \( G \). This index is given by the formula:\[ [G:H] = \frac{|G|}{|H|} \]where |G| and |H| are the orders of groups \( G \) and \( H \), respectively.In this exercise, we consider two indices: \([G:H] = n\) and \([H:K] = m\). We show that multiplying these indices results in the index \([G:K] = n \times m\). This multiplication reflects how the subgroup \( K \) interacts within the broader group contexts of \( H \) and \( G \), helping us understand the multi-step hierarchy from \( G \) to \( K \). Consequently, this index gives us the total number of cosets that \( K \) forms in \( G \). This understanding is crucial for simplifying group-based problems, as it provides a clear path across different subgroup levels.