Problem 33

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 $G$.)

Step-by-Step Solution

Verified

Answer

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

1Step 1: 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 $H$.

2Step 2: 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.

3Step 3: 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 $x_i y_j K$.

4Step 4: 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

SubgroupFinite GroupCosetsIndex of a Subgroup

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.

Problem 33

Problem 34

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Problem 33

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