Problem 1 - A Book of Abstract Algebra

Question

In each of the following, show that \(H\) is a subgroup of \(G\). \(G=\langle F(\mathbb{F}),+\rangle, H=\\{f \in \mathscr{F}(\mathbb{R}): f(x)=0\) for every \(x \in[0,1]\\}\)

Step-by-Step Solution

Answer

\(H\) is a subgroup of \(G\) because it satisfies identity, closure, and inverse properties.

Step 1: Understanding the Groups

First, we need to clarify what sets we are dealing with. Given that \(G = \langle F(\mathbb{F}), +\rangle\), we understand it as the group of all functions from a field \(\mathbb{F}\) with pointwise addition. The subset \(H\) consists of all real-valued functions defined on \([0,1]\) that are identically zero on that interval.

Step 2: Check the Identity Element Condition

A subgroup \(H\) must contain the identity element of \(G\). In \(G\), the identity element is the zero function, i.e., a function \(f\) such that \(f(x) = 0\) for all \(x\). Since each function in \(H\) satisfies \(f(x) = 0\) for each \(x \, \in \,[0,1]\), \(H\) contains the identity element.

Step 3: Check Closure Under the Group Operation

For closure under addition, for any two functions \(f, g \in H\), consider their sum \((f+g)(x) = f(x) + g(x)\). Because both \(f(x) = 0\) and \(g(x) = 0\) for all \(x \in [0,1]\), \((f+g)(x) = 0\) for these \(x\). Thus, \(f+g \in H\), satisfying closure under addition.

Step 4: Check the Inverse Condition

For the subgroup property, each element \(f \in H\) must have an inverse also in \(H\). The inverse under addition is \(-f\). Since \(f(x) = 0\) for all \(x \in [0,1]\), \((-f)(x) = -0 = 0\) for these \(x\). Thus, \(-f \in H\), confirming the inverse condition is satisfied.

Step 5: Conclusion of Subgroup Verification

Since \(H\) contains the identity, is closed under addition, and contains an inverse for each of its elements, \(H\) satisfies the subgroup criteria. Thus, \(H\) is a subgroup of \(G\).

Key Concepts

Group Theory

In group theory, the concept of a subgroup plays a crucial role. At its core, a group is a set combined with an operation that meets four key criteria:

• Closure: Stays within the set when applying the operation.
• Associativity: Groups elements properly irrespective of brackets.
• Identity Element: An element that doesn't change others when used with the operation.
• Inverse Element: Every element has a counterpart that returns it to the identity element.

A subgroup is a smaller set within a group that retains all these properties. Subgroups are essential because they allow us to examine the structure of complex groups in simpler terms.
They let us break down larger symmetries into more manageable parts.

Identity Element

The identity element is a fundamental part of any group structure. It is the element in a group that does nothing when combined with any other element. In mathematical terms, for a group operation "6", an identity element "e" satisfies the equation:

• 6(e, a) = a
• 6(a, e) = a

for any element "a" in the group.
In our context, where the group operation is addition of functions, the identity element is the zero function. This means that for any function "f", when you add it to the zero function, it remains unchanged. If "f" is in the subgroup "H", it naturally meets this requirement because they are already zero functions over the interval [0,1]. This criterion is pivotal as it confirms "H" includes the identity element of "G".

Closure Property

The closure property ensures that performing the group operation on any two elements of a subgroup results in another element that is also in that subgroup. For "H" to be a subgroup of "G", it must be closed under the addition of functions.
If you take any two functions in "H", say "f" and "g", known to be zero over [0,1], their sum must also yield a function in "H". Mathematically, this translates to:

• 6(f, g)(x) = f(x) + g(x) = 0 + 0 = 0

for all "x" in [0,1].
This property confirms that within "H", combining any two functions keeps you within the constraints of the subgroup. It supports "H" being closed under the group operation, adhering to the subgroup definition within "G".

Inverse Element

For a set to be a subgroup, each of its elements must have an inverse that also belongs to the set. This assures that whatever element you start with, you can always find another in the set to return to the identity element.
In our group defined by function addition, the inverse of any function "f" is denoted by "-f". For a function that is zero everywhere on [0,1], "-f" will also be zero over that interval. This is because negating zero still results in zero. Symbolically, this means:

• 6(-f, f)(x) = -f(x) + f(x) = 0

for all "x" in [0,1].
This condition meets the criteria for inverses, confirming that each zero function has its negation (also a zero function) within "H". Thus, "H" satisfies the requirement for inverse elements in its set, allowing "H" to be classified as a subgroup of "G".