Problem 4 - A Book of Abstract Algebra

Question

Let \(G\) be a group; let \(H\) and \(K\) be subgroups of \(G\), with \(H\) a normal subgroup of \(G\). Prove the following : Every member of the quotient group \(H K / H\) may be written in the form \(H k\) for some \(k \in K\).

Step-by-Step Solution

Answer

Each member of \(HK/H\) is of the form \(Hk\), with \(k \in K\).

Step 1: Understanding the Quotient Group

The quotient group \(HK/H\) is defined by the cosets of \(H\) in \(HK\). So, each element of \(HK/H\) is of the form \(H(hk)\) for some \(hk \in HK\).

Step 2: Expression of Element in Quotient

Any element \(hk\) in \(HK\) can be expressed with \(h \in H\) and \(k \in K\), so any coset in \(HK/H\) is \(H(hk) = Hk\) because \(Hh = H\) since \(h \in H\). Here we are using the normality of \(H\) to claim \(Hh = H\).

Step 3: Conclusion

Therefore, any element of the quotient group \(HK/H\) can be written in the form \(Hk\), where \(k \in K\). This shows that every coset of \(H\) in \(HK\) s implied to have a representative solely from \(K\).

Key Concepts

Quotient Group

In group theory, a quotient group, also known as a factor group, is a type of group formed by partitioning a group into disjoint subsets called cosets. These cosets stem from the subgroup within the original group, which is denoted as a normal subgroup. The resulting structure is itself a group, termed the quotient group.
Understanding quotient groups begins with the concept of a subgroup. Imagine a group, \(G\), and a normal subgroup, \(H\). We can form the quotient group \(G/H\) by considering all possible left cosets of \(H\) in \(G\), which are the subsets \(gH\) for each element \(g\) in \(G\).

• Quotient groups simplify a group by identifying and collapsing similar elements that differ only by an element of the subgroup \(H\).
• The operation of the quotient group involves combining these cosets in a way that respects the original group operation.

Normal Subgroup

A normal subgroup is a subgroup that is invariant under conjugation by elements of the group it belongs to. This property makes it possible to construct the quotient group.
The defining characteristic of a normal subgroup \(H\) in a group \(G\) is that for every element \(g\) in \(G\), the equation \(gHg^{-1} = H\) holds. Essentially, the sequence of applying the inverse and then the element does not change \(H\).

• Normal subgroups allow consistent definition of cosets and ensure the group operations are well-defined on the quotient.
• The concept of normality is pivotal in ensuring that the algebraic structure of a quotient group behaves properly.

Consequently, any elements generated in a quotient group are combinations of elements from a subgroup \(H\) and elements of another subgroup \(K\), as seen in the exercise above. This concept helps affirm that each coset of \(H\) in \(HK/H\) can be expressed simply as \(Hk\) for some element \(k\) in \(K\).

Subgroup

A subgroup is a subset of a group that also forms a group under the same operation. For any subgroup \(H\), it must satisfy certain criteria: it needs to have the group identity element, inverses of its elements, and be closed under the operation of the larger group \(G\).
Subgroups play a vital role in forming cosets and subsequently quotient groups.
In the exercise, both \(H\) and \(K\) are subgroups of \(G\). The intersection of these groups forms the basis for building further structures like \(HK\), illustrating the versatility of subgroup concepts in group theory.

• Subgroups form a foundation for more complex structures, acting as building blocks within a group.
• They allow for an exploration of properties within larger mathematical entities, hence their importance in quotient group formation.

Cosets

Cosets arise when you partition a group \(G\) by a subgroup \(H\), effectively forming smaller, non-overlapping subsets. Each coset is of the form \(gH\) or \(Hg\), containing all elements obtained by multiplying each element of \(H\) by a specific group element \(g\).
These cosets are crucial in defining quotient groups. In the context of the exercise, we consider the cosets of \(H\) in \(HK\), forming the elements of the quotient group \(HK/H\).

• Left cosets (\(gH\)) and right cosets (\(Hg\)) are identical when \(H\) is a normal subgroup, simplifying the formation of the quotient group.
• Cosets do not overlap, each contains a distinct collection of group elements, which aids in building the structure of the quotient group efficiently.

By focusing on cosets, we derive elements of a quotient group, ensuring these elements uphold the group properties due to the normality of \(H\). This understanding allows us to express elements like \(Hk\) in \(HK/H\), leading to straightforward expression of the group structure in terms of representative elements from \(K\).