Question

Show that the alternating group \(A_{4}\) contains a subgroup isomorphic to the Klein 4-group \(V\).

Short answer

The subgroup \( \{ e, (12)(34), (13)(24), (14)(23) \} \) in \(A_4\) is isomorphic to the Klein 4-group \(V\).

Step-by-step solution

Understand Alternating Group A_4

The alternating group \(A_4\) is the group of even permutations of four elements. It has 12 elements and is a subgroup of the symmetric group \(S_4\).

Define the Klein 4-group V

The Klein 4-group \(V\), also denoted by \(V_4\), is a group with four elements, specifically \( \{ e, a, b, c \} \), where \(e\) is the identity, and the operations are such that every element is its own inverse and the multiplication of any two distinct elements is the third element.

Identify Elements of A_4 Equivalent to V

In \(A_4\), seek elements that can mimic the Klein 4-group structure. Notice that the set of permutations \( \{ e, (12)(34), (13)(24), (14)(23) \} \) has four elements like \(V_4\).

Verify Subgroup Properties

Check that \{ e, (12)(34), (13)(24), (14)(23) \} is closed under multiplication and every element is its own inverse. Verify that each element squares to the identity and multiplication of any two distinct elements yields the third, confirming that it is a subgroup and maps isomorphically to \(V\).

Confirm Group Closure and Structure

Perform the multiplication: - \((12)(34) \circ (12)(34) = e\)- \((12)(34) \circ (13)(24) = (14)(23)\)- \((12)(34) \circ (14)(23) = (13)(24)\)- Other combinations similarly affirm group closure and match with \(V\)'s operation table.

Key concepts

Alternating Group A4

The alternating group, denoted as \(A_4\), is a fundamental concept in group theory. It consists of all even permutations of four distinct elements. To understand what an even permutation is, it helps to imagine rearranging items. If this rearrangement can be achieved by making an even number of swaps, it is considered an even permutation.
In total, \(A_4\) contains 12 elements, making it a subgroup of the symmetric group \(S_4\), which consists of all permutations of four elements. Within this group, each element is a unique rearrangement that involves an even number of transpositions, or swaps.
Understanding the structure of \(A_4\) is crucial because it helps in analyzing symmetry in various mathematical and physical contexts.

Group Isomorphism

Group isomorphism is a significant concept in abstract algebra. It describes a relationship between two groups that are structurally identical, or "isomorphic." In essence, it means there exists a one-to-one correspondence, or bijection, between the groups that preserves the group operation.
If two groups \(G\) and \(H\) are isomorphic, denoted by \(G \cong H\), they share the same structure. This implies that the way elements combine in \(G\) mirrors exactly how elements combine in \(H\).
Recognizing group isomorphism is crucial because it allows mathematicians to consider different groups as essentially the same, allowing for the application of established properties and theorems to new contexts without re-proving them for each specific group.

Permutation Group

A permutation group is a collection of permutations that form a group. A permutation itself is a specific way of arranging or ordering a set of elements. For mathematicians, permutation groups are all about symmetry, determining how you can rearrange elements of a set while maintaining some structure.
Each permutation can be seen as a function that reorders elements. If you have a set of elements, the symmetric group \(S_n\) of them includes every possible way to rearrange those elements.
Permutation groups are foundational in group theory because they provide a concrete and intuitive example of abstract group concepts. They serve as the playground for understanding more complex algebraic structures and their symmetries.

Subgroup

In group theory, a subgroup is a subset of a group that is itself a group under the operation of the larger group. For a set \(H\) to be a subgroup of \(G\), it must satisfy three criteria: it must include the identity element of \(G\), it must be closed under the group operation, and every element in \(H\) must have an inverse in \(H\).
Subgroups are important because they reveal the internal structure of groups. Discovering a subgroup often means uncovering deeper symmetries and patterns within the larger group.
When a problem involves showing one group as a subgroup of another, it often involves demonstrating these criteria are met with specific elements that mirror similar characteristics, as is the case with \(A_4\) and the Klein 4-group.

Even Permutations

Even permutations are key to understanding symmetrical arrangements within a group of elements. An even permutation on a set is a permutation that can be decomposed into an even number of transpositions (element swaps). For example, swapping elements twice yields an even permutation.
In the context of the alternating group \(A_4\), all elements are even permutations, which gives the group its even nature. Identifying even permutations helps isolate certain subgroup characteristics and play a role in proving isomorphisms between groups.
Focusing on even permutations allows mathematicians to explore the rich field of symmetries further, which is crucial in fields like physics and chemistry where understanding molecular structures is essential.