Question

Let \(H\) and \(K\) be subgroups of a finite group \(G\) with index \([G: H]=n\) and index \([G: K]=m .\) Show that \(\operatorname{lcm}(n, m) \leq[G: H \cap K] \leq n m .\)

Short answer

\(\operatorname{lcm}(n, m) \leq [G: H \cap K] \leq nm\)

Step-by-step solution

Understand the Given Information

We have a finite group \(G\) and two subgroups \(H\) and \(K\). The indices \([G: H] = n\) and \([G: K] = m\) indicate the number of distinct left cosets of \(H\) and \(K\) in \(G\), respectively.

Analyze the Intersection of Subgroups

The subgroup intersection \(H \cap K\) consists of elements that are in both \(H\) and \(K\). We need to find the index \([G : H \cap K]\). This is the number of distinct left cosets of \(H \cap K\) in \(G\).

Use the Relationship of Indices

To show the inequalities, utilize the formula:\[[G : H \cap K] = \frac{[G : H][G : K]}{[H : H \cap K][K : H \cap K]}.\]Since indices are positive integers, \([H : H \cap K] \geq 1\) and \([K : H \cap K] \geq 1\), which implies:\[[G : H \cap K] \leq [G : H][G : K] = nm.\]

Lower Bound - Least Common Multiple

The lower bound is established by showing that \([G : H \cap K]\) is at least the least common multiple of \(n\) and \(m\). Since any left coset of \(H\) must contain at least \(n\) distinct cosets of \(H \cap K\), and similarly for \(K\), it follows:\[[G : H \cap K] \geq \operatorname{lcm}(n, m).\]

Conclusion of the Inequality

Combining the results from the previous steps, we have established:\[\operatorname{lcm}(n, m) \leq [G : H \cap K] \leq n m.\]

Key concepts

Subgroups

In the world of group theory, a subgroup is a smaller group contained within a larger group. Think of it like a club that has rules similar to the bigger group they are part of, but with less members. Formally, if you have a group \(G\), a subgroup \(H\) is a subset of \(G\) such that all elements of \(H\) behave under the group's operation like the elements of \(G\) itself.

• Every subgroup must include the identity element of the main group since this element leaves other elements unchanged.
• Subgroups must be closed under the group's operation, which means combining any two elements in the subgroup should also result in an element that's in the subgroup.
• Inverses are part of subgroups, meaning for every element in the subgroup, its inverse must also be in the subgroup.

Subgroups are foundational because they maintain the order and structure of the larger group \(G\) while giving insights into its composition.

Cosets

Cosets are a way to break down groups into simpler parts. If you have a subgroup \(H\) from the group \(G\), you can think about how elements from \(G\) combine with \(H\). Each unique combination results in a coset. It's similar to how you might combine various ingredients to make different dishes from the same recipes.

• The left coset of an element \(g\) in \(G\) with respect to \(H\) is the set \(gH = \{gh : h \in H\}\).
• Cosets partition the group \(G\) into non-overlapping subsets, meaning you can divide \(G\) into complete cosets formed by \(H\).
• No two different cosets share elements; they are distinct from each other.

Understanding cosets is essential for analyzing the internal structure of groups and helps in understanding indices and other group properties.

Index of a Group

The index of a subgroup \(H\) in a group \(G\), written as \([G:H]\), indicates the number of cosets formed when \(G\) is divided by \(H\). You can think of this as how many different ways you can rearrange \(H\) to recreate \(G\) through cosets.

• Index is calculated by counting the distinct cosets of \(H\) in \(G\).
• This number reflects how 'big' \(G\) is in relation to \(H\).
• For finite groups, the index \([G:H]\) is always a whole number, because it counts these clearly defined partitions.

The concept of index helps to understand the proportion of the subgroup within the group, and is crucial when analyzing the intersection with other subgroups, as seen in problems like the one we are discussing.

Least Common Multiple (LCM)

The least common multiple, often abbreviated as LCM, is the smallest number that two or more numbers share as a multiple. It’s like finding the smallest pot where all of your different soup ingredients can fit perfectly without leftovers.

• The LCM of two numbers \(n\) and \(m\) can be found by listing the multiples of the numbers and identifying the smallest common one.
• For example, the LCM of 4 and 5 is 20 because 20 is the first multiple that both numbers divide evenly into.
• In group theory, LCM helps find shared patterns or overlaps in group structures, such as when examining indices or subgroup interactions.

In the context of subgroups and index, LCM is used to establish bounds in inequalities, highlighting the relationship between different subgroups in a group.