Problem 19 - Algebra: Pure and Applied

Question

Give at least two examples of a nontrivial proper subgroup of the indicated group. $$ \mathrm{Q}_{8} $$

Step-by-Step Solution

Answer

The subgroups \( \langle i \rangle = \{ 1, i, -1, -i \} \) and \( \langle j \rangle = \{ 1, j, -1, -j \} \) are examples of proper subgroups of \( \mathrm{Q}_8 \).

Step 1: Understand the Quaternion Group \( \mathrm{Q}_8 \)

The quaternion group \( \mathrm{Q}_8 \) consists of the elements \( \{ 1, -1, i, -i, j, -j, k, -k \} \). It is a non-abelian group of order 8, meaning there are 8 elements. The group operation is multiplication, where \( i^2 = j^2 = k^2 = ijk = -1 \). It is a common example of a non-commutative group.

Step 2: Definition of Subgroup and Proper Subgroup

A subgroup \( H \) of a group \( G \) is a subset of elements in \( G \) that is itself a group under the operation defined on \( G \). A proper subgroup is a subgroup that is not equal to the entire group \( G \). For \( \mathrm{Q}_8 \), we want subgroups other than \( \mathrm{Q}_8 \) itself or the trivial group \( \{1\} \).

Step 3: Example 1: Subgroup Generated by \( i \)

Consider the subgroup \( \langle i \rangle = \{ 1, i, -1, -i \} \). This subgroup is generated by \( i \), since \( i^2 = -1 \), \( i^3 = -i \), and \( i^4 = 1 \). Thus, \( \langle i \rangle \) is a proper subgroup of order 4.

Step 4: Example 2: Subgroup Generated by \( j \)

Consider the subgroup \( \langle j \rangle = \{ 1, j, -1, -j \} \). This subgroup is generated by \( j \), since \( j^2 = -1 \), \( j^3 = -j \), and \( j^4 = 1 \). Thus, \( \langle j \rangle \) forms another proper subgroup of order 4.

Step 5: Verification of Properness

Both \( \langle i \rangle \) and \( \langle j \rangle \) are proper because they do not contain all elements of \( \mathrm{Q}_8 \), specifically lacking elements like \( k \) and \( -k \). This ensures they are indeed smaller subgroups and hence proper.

Key Concepts

Non-abelian group

A non-abelian group is a type of group where the order in which you perform operations matters. In simpler terms, for a non-abelian group, it's possible to find at least two elements, let's call them \( a \) and \( b \), for which \( ab eq ba \). This property distinguishes non-abelian groups from abelian groups, where every pair of elements commutes, meaning \( ab = ba \) for all elements \( a \) and \( b \).

The quaternion group \( \mathrm{Q}_8 \) is an example of a non-abelian group. With 8 elements, this group uses multiplication as its operation. Due to certain combinations of elements like \( i, j, \) and \( k \), the operations do not commute. For instance, \( ij = k \) but \( ji = -k \); hence the non-abelian nature. Understanding these groups is essential in fields like theoretical physics, where symmetries play a crucial role.

Proper subgroup

A proper subgroup is a smaller collection of elements within a greater group that forms its own group under the same operation as the larger group. Importantly, a proper subgroup is neither the whole group itself nor merely the trivial subgroup, which only includes the identity element.

To clarify, in any group \( G \), the sets \( G \) and \( \{e\} \) (where \( e \) is the identity element) are known as trivial subgroups, because they include either all elements or just the identity. A **proper** subgroup is something more interesting. For example, in \( \mathrm{Q}_8 \), a subgroup like \( \langle i \rangle = \{ 1, i, -1, -i \} \) is a proper subgroup since it contains more than the singular identity but less than the full set of \( \mathrm{Q}_8 \).

Group elements

In the context of group theory, elements refer to the individual components or "members" of a group. These elements interact with each other according to specific rules dictated by the group operation. For instance, the quaternion group \( \mathrm{Q}_8 \) includes elements such as \( 1, -1, i, -i, j, -j, k, -k \).

Each of these elements has unique properties and relationships within the group. Take \( i, j, \) and \( k \) – these aren't just symbols; they interact through defined rules: \( i^2 = j^2 = k^2 = ijk = -1 \). This means squaring any of these elements results in \( -1 \), and their specific order in this product results in \( -1 \), showcasing the intricate functionality of group elements. Getting to know these elements helps in understanding how groups like \( \mathrm{Q}_8 \) function as a whole.

Group order

The concept of group order refers to the total number of elements within a group. It is a fundamental characteristic that influences the group's structure and properties. For the quaternion group \( \mathrm{Q}_8 \), the order is 8, indicating there are 8 distinct elements.

Recognizing a group's order helps in analyzing properties like subgroups and defining relationships between elements. The order also plays a critical role in determining the presence of proper subgroups. Subgroups will have orders that are divisors of the group's order. In \( \mathrm{Q}_8 \), prospective subgroups could have orders like 1, 2, or 4. Proper subgroups such as \( \langle i \rangle \) and \( \langle j \rangle \) have an order of 4, correctly dividing the total group order. This foundational knowledge aids in navigating more complex group characteristics.