Question

Let \(H\) be a subgroup of a group \(G\). Show that for any \(a \in G\) we have \(|H a|=|H|\).

Short answer

The sets \( Ha \) and \( H \) have the same cardinality since there is a bijection between them.

Step-by-step solution

Understanding the Problem

We have a subgroup \( H \) of a group \( G \) and we need to show that the set \( Ha = \{ ha : h \in H \} \) has the same cardinality (or size) as \( H \) for any element \( a \) in \( G \). This means showing \( |Ha| = |H| \).

Define a Function Between Sets

Let's define a function \( f: H \rightarrow Ha \) by \( f(h) = ha \) for each \( h \in H \). This function maps every element \( h \) in \( H \) to an element in \( Ha \) by multiplying \( h \) on the right by \( a \).

Show the Function is Well-Defined

Since \( a \) is fixed, for each \( h \in H \), \( ha \) is a definite element in \( Ha \) because \( Ha \) is defined as the set of all \( ha \) where \( h \in H \). Thus, \( f \) is well-defined.

Show the Function is Injective

To show \( f \) is injective, assume \( f(h_1) = f(h_2) \) for \( h_1, h_2 \in H \). Then, \( h_1a = h_2a \). By cancelling \( a \) (as \( G \) is a group and every element has an inverse), we obtain \( h_1 = h_2 \) which shows that \( f \) is injective.

Show the Function is Surjective

To show \( f \) is surjective, take any element \( ha \in Ha \). We want to show there exists some \( h' \in H \) such that \( f(h') = ha \). By definition of \( Ha \), there exists an \( h \in H \) such that \( ha \) is already in the image of \( f \). Thus, \( f \) is surjective.

Conclude

Since the function \( f: H \rightarrow Ha \) is both injective and surjective, it is a bijection. Thus, \( |Ha| = |H| \).

Key concepts

Subgroups

In group theory, a **subgroup** is a subset of a group that itself forms a group under the same operation as the larger group. If you know about set theory, think of it as a special subset with its own distinct identity. For a subset to be a subgroup, it must satisfy certain properties. It should contain the identity element of the original group, be closed under the group operation, and for every element in this subset, its inverse should also be present.

When talking about subgroups, let's use a simple analogy. Imagine a country called Group-Land, where everyone speaks the language "Math." Now, a subgroup is like a neighborhood in this country where all the citizens are bound by the same rules and language, but they might have their own dialect. You can visit this neighborhood without stepping out of Math-speaking environments.

To decide if a subset qualifies as a subgroup, remember:

• The identity of the main group should also be in the subgroup.
• Any operation between two elements of the subgroup still lands you in the subgroup.
• Each element in the subgroup should have its inverse present in the subgroup as well.

Exploring subgroups can give insights into the structure and characteristics of the larger group, much like learning about neighborhoods tells you more about the overall culture of Group-Land.

Group Homomorphism

A **group homomorphism** is a bridge between two groups, a function that connects the structure of one group to another. When working with groups, maintaining their algebraic structure while transforming elements from one to another is crucial. Homomorphisms do precisely this by preserving the group operations during the transformation.

Think about it like translating sentences from one language to another, where the meaning remains intact despite different words being used. A group homomorphism is a function \( f: G \rightarrow K \), where for any two elements \( a, b \in G \), it satisfies

• \( f(ab) = f(a)f(b) \)

This ensures that the structure of the group’s operations is preserved. Homomorphisms keep the essence of a group’s operations intact through transformation, allowing us to compare the internal structures of different groups.

Group homomorphisms play a foundational role in understanding broader algebraic systems, helping researchers compare how different groups relate to each other, much like understanding language roots can reveal connections between cultures.

Bijective Functions

**Bijective functions** are a type of function that make a perfect "match" between elements of two sets. They are functions that are both injective (one-to-one) and surjective (onto). In simple terms, a bijective function ensures that every element in the first set has exactly one unique partner in the second set, and every element of the second set is matched up with an element from the first.

Imagine a school where each student must have exactly one desk, and every desk must have one student. This one-to-one and onto relationship ensures there is a perfect pairing without any empty desks or homeless students.

For a function \( f: A \rightarrow B \):

• An injective function signals that different elements from set A map to different elements in set B.
• A surjective function ensures each element of set B is the image of at least one element of set A.
• When a function is both, it becomes bijective, creating an ideal correspondence between all elements of the two sets.

Bijective functions, therefore, are central to demonstrating equivalences between mathematical structures, ensuring each part of one structure is perfectly paired with a part of another.

Cardinality in Groups

The concept of **cardinality** essentially refers to the "size" or "number of elements" of a set. When it comes to groups, cardinality plays a critical role in understanding and comparing different groups.

The idea of cardinality helps us determine not just whether two groups are equivalent, but to what extent. For instance, a subgroup of a group will certainly have a cardinality less than or equal to the cardinality of the original group because it cannot have more elements than the group it belongs to.

When comparing groups, one of the primary tasks is often to determine if there exist bijective functions between them, which can indicate that they have the same cardinality.

• For finite groups, cardinality is simply the number of elements in the group.
• For infinite groups, this involves more abstract thinking, like counting different "infinities" as done with real numbers versus natural numbers.

Understanding cardinality is crucial for group theory as it supports the foundational understanding that allows us to navigate between subgroups, supergroups, and other related algebraic systems.