Question

Establish a bijection between the set of right cosets and the set of left cosets of a subgroup. Hint: Define a map that takes \(S t\) to \(t^{-1} S\).

Short answer

The function \(f: St \to t^{-1}S\) provides a bijective map between the set of right cosets and the set of left cosets of a subgroup.

Step-by-step solution

Definition of Cosets

Let's start by defining what a coset is. Given a subgroup \(H\) of a group \(G\), a right coset of \(H\) in \(G\) is a set of the form \(Ht\), where \(t \in G\). Similarly, a left coset of \(H\) in \(G\) is of the form \(tH\), where \(t \in G\).

Defining the Map

We define a map \(f\) as follows: for any right coset \(St\), where \(S\) is our subgroup and \(t\) is an element of our group, we map \(St\) to \(t^{-1}S\). This is a map from the set of right cosets to the set of left cosets.

Proving Injectivity

Next, we prove that \(f\) is injective. To do that, we must show that if \(St = Sv\) for some \(t, v \in G\), then their images under \(f\) are also equal. Observe that if \(St=Sv\), then \(t^{-1}S = v^{-1}S\), which demonstrates the injectivity of \(f\).

Proving Surjectivity

Lastly, we have to prove that \(f\) is surjective. This means for any left coset \(tS\), \(t \in G\), there exists a right coset \(Sv\), for some \(v \in G\), such that \(f(Sv) = tS\). We can simply select \(Sv = St^{-1}\). Then \(f(St^{-1}) = (t^{-1})^{-1}S = tS\), showing surjectivity of \(f\).

Key concepts

Cosets

In group theory, cosets play a fundamental role in understanding the structure of groups as they relate to their subgroups. A coset can be thought of as a way of "shifting" a subgroup across the elements of a group. Specifically, if you have a subgroup \(H\) within a group \(G\), you can form either a right coset or a left coset.

• A right coset of \(H\) in \(G\) is written as \(Ht\), where \(t\) is an element in \(G\). This means you take each element of \(H\) and multiply it on the right by \(t\).
• A left coset is expressed as \(tH\), wherein you multiply \(t\) on the left side of each element in \(H\).

Cosets partition the group into equal-sized, disjoint sets that cover the entire group. They serve as a tool for analyzing whether two groups are isomorphic and help in studying the factor groups, which are vital in understanding normal subgroups.

Bijective Maps

Bijective maps are essential in mathematics because they establish a one-to-one correspondence between two sets, implying both a perfect pairing and a match in size between the sets. A map is bijective if it is both injective and surjective.To prove a map is injective, you show that different elements in the domain map to different elements in the codomain. For the map \(f\) between the right cosets and left cosets, if \(St = Sv\) implies \(t^{-1}S = v^{-1}S\), then the map is injective.For a map to be surjective, every element in the codomain must have a pre-image in the domain. In the example, for any left coset \(tS\), there exists a right coset \(Sv\) such that \(f(Sv) = tS\), proving that \(f\) is surjective.Thus, bijective maps help bridge the structure of different algebraic sets, enabling deeper insights into group properties.

Subgroups

Subgroups are the building blocks of a group in algebra, much like atoms are to molecules. If you have a group \(G\), a subgroup \(H\) is a smaller group within \(G\) that itself satisfies the group axioms: closure, associativity, identity, and invertibility.

• Closure: If \(a, b \in H\), then \(ab \in H\).
• Associativity: The operation in \(H\) satisfies \((ab)c = a(bc)\).
• Identity: There exists an element \(e \in H\) such that for all elements \(a \in H\), \(ea = ae = a\).
• Invertibility: For every element \(a \in H\), there exists an element \(b \in H\) such that \(ab = ba = e\).

Subgroups allow us to decompose complex groups into simpler, more manageable pieces. They also help define other concepts like cosine, normal subgroups, and group homomorphisms, each of which plays a significant role in unraveling the complexities of algebraic structures.