Problem 1 - A Course in Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry

Question

(a) Show that If \(H\) and \(K\) are subgroups of \(G\) then therr mtersection \(H \cap K\) is always a subgroup of \(G\) (b) Show that the product \(H K=[h k \mid h \in H, k \in K]\) is a subgroup if and only if \(H K=K H\)

Step-by-Step Solution

Answer

The intersection \(H \cap K\) of any two subgroups \(H\) and \(K\) of a group \(G\) is always a subgroup of \(G\). The product \(HK\) of the same subgroups is a subgroup if and only if \(HK = KH\).

Step 1: Prove intersection of subgroups

Subgroups \(H\) and \(K\) are subsets of group G satisfying three properties of subgroup. Let \(x, y \in H \cap K\), so they are also in both \(H\) and \(K\). \n(i) Closure property: Since \(H\) and \(K\) are subgroups, the operation of their elements produces another element in them. Hence, \(x*y \in H\) and \(x*y \in K\) so \(x*y \in H \cap K\). \n(ii) Identity property: Both \(H\) and \(K\) contain the group \(G\) identity, so it belongs to \(H \cap K\) too. \n(iii) Inverse property: The inverses of \(x (x^{-1})\) are in \(H\) and \(K\), therefore are in \(H \cap K\). Hence, \(H \cap K\) satisfies all subgroup criteria and is a subgroup of \(G\).

Step 2: Prove the product \(HK\) is a subgroup

For this, we need to show that \(HK\) satisfies the properties to be a subgroup of \(G\) given \(HK = KH\). \n(i) Closure property: Consider two elements \(h_1k_1, h_2k_2 \in HK\). Their product is \(h_1k_1(h_2k_2)=h_1(k_1h_2)k_2\) (since \(HK = KH\), we can write \(k_1h_2\) instead of \(h_2k_1\)). Let \(h_3=k_1h_2 \in HK\), then \(k_1h_2k_2 \in HK\) and closure property holds. \n(ii) Identity property: Both \(H\) and \(K\) contain the group \(G\) identity, hence their product will also contain the group identity. \n(iii) Inverse property: Consider an element \(hk \in HK\). Its inverse \(k^{-1}h^{-1}\) (since \(HK = KH\), this pairs with its group product member) is also \( \in HK\). So, \(HK\) indeed forms a subgroup of \(G\) when \(HK = KH\).

Step 3: Prove product \(HK\) is not a subgroup if \(HK \neq KH\)

Assume \(HK \neq KH\). Then, exist \(h_1, h_2 \in H, k_1, k_2 \in K\) such that \(h_1k_1 \neq h_2k_2\) which violates the closure property and hence, \(HK\) is not a subgroup.

Key Concepts

Subgroups

In group theory, a subgroup is a special concept that appears frequently. A subgroup is simply a smaller group within a larger group. For instance, if we have a group \(G\), a subset \(H\) of \(G\) is a subgroup if it itself can operate as a group. This means \(H\) must also follow the same rules that \(G\) does. For \(H\) to be a subgroup, it must satisfy three properties: closure, identity, and inverse. Subgroups can help us determine a lot about the larger group by breaking it down into smaller parts. Understanding how these subgroups interact with each other, such as their intersection, is crucial to deeper group theory insights. In particular, the intersection of two subgroups \(H\) and \(K\), noted as \(H \cap K\), will also be a subgroup. This is proven by ensuring that their intersection adheres to the fundamental subgroup properties.

Closure Property

The closure property is a fundamental concept in group theory and is crucial for determining whether a set forms a subgroup. This property states that for any two elements, \(a\) and \(b\), in the group, their operation (such as addition or multiplication) should still be within the group. In simpler terms, if you take two elements from the set and 'combine' them, the result should also be an element of the same set.

For example, if \(x, y \in H\) for subgroups \(H\), then their product \(xy\) should also be an element of \(H\). Ensuring the closure property holds is essential for both proving the intersection of subgroups and for proving products of subgroups (like \(HK\)) form subgroups under certain conditions. This property emphasizes the idea that a subgroup should always contain the operation results of its elements, making it a complete subset under the group's operation.

Identity Property

The identity property is another crucial aspect of group and subgroup structures. The identity element in a group is a special element, which, when combined with any element of the group, leaves the other element unchanged. For instance, in a group under addition, the identity element would be \(0\); in a group under multiplication, it would be \(1\).

In order for a subgroup \(H\) within a group \(G\) to be valid, the identity element of \(G\) must also be present in \(H\). This means that in the intersection of two subgroups, \(H \cap K\), the identity element must be there because it belongs to both \(H\) and \(K\). For products of subgroups, such as \(HK\), having the identity element ensures the completeness of the subgroup's structure, reinforcing the group's internal consistency.

Inverse Property

The inverse property is essential for any group or subgroup. It is the idea that for every element in the group, there should exist another element, known as its inverse, that can combine with the original to return the identity element.

For instance, for an element \(x\), its inverse \(x^{-1}\) satisfies the equation \(xx^{-1} = e\), where \(e\) is the identity element of the group. In a subgroup \(H\), every element must have its inverse also contained in \(H\). This criterion is vital when considering the intersection of subgroups, as well as in proving the properties of products of subgroups, such as \(HK\). Without the presence of inverses, the subgroup would lack the ability to 'undo' operations, which is a definitive aspect of a complete group structure. The inverse property ensures reversibility within group operations, contributing to the mathematical elegance and functionality of groups and subgroups.