Klein four-group
The Klein four-group, denoted V_4 or \mathbb{Z}_2 \times \mathbb{Z}_2, is the unique non-cyclic abelian group of order four, consisting of an identity element and three elements each of order two, where the product of any two distinct non-identity elements is the third.[1] It is isomorphic to the direct product of two cyclic groups of order two and serves as a fundamental example of an elementary abelian 2-group in abstract algebra.[2] Every element in the group is its own inverse, and its Cayley table reflects this commutative structure, with the group operation yielding the identity when any element is composed with itself.[3] Named after the German mathematician Felix Klein, who highlighted its properties during his investigations into icosahedral symmetries and the solution of polynomial equations in the late 19th century, the group emerged as a key structure in early group theory.[4] Klein discussed it in lectures as the smallest non-cyclic group, underscoring its role in unifying geometric and algebraic concepts through his Erlangen Program.[4] Historically, it predates formal naming but gained prominence in Klein's work on non-Euclidean geometry and function theory around 1870–1880.[4] The Klein four-group exhibits several notable properties, including being nilpotent, solvable, and supersolvable, with an automorphism group isomorphic to the symmetric group S_3.[2] It contains exactly three proper nontrivial subgroups, each cyclic of order two, all normal due to its abelian nature.[1] In applications, it arises as the symmetry group of a rectangle (or a non-square rhombus), capturing rotations by 180 degrees and reflections over the axes of symmetry.[1] Additionally, it forms the additive group of the finite field with four elements and embeds naturally in larger groups like the alternating group A_4 or the dihedral group of order eight, making it essential for studying classifications of small finite groups and modular representations.[2]Definition and Properties
Definition
The Klein four-group is the unique (up to isomorphism) abelian group of order four that is not cyclic. It is commonly denoted by V_4 (from the German Vierergruppe, meaning "four-group") or K_4.[5] The group consists of an identity element e and three non-identity elements a, b, and c, each satisfying a^2 = b^2 = c^2 = e.[6] Since the group is abelian, its multiplication table is symmetric, with products defined by ab = ba = c, ac = ca = b, and bc = cb = a. The full table is:| \cdot | e | a | b | c |
|---|---|---|---|---|
| e | e | a | b | c |
| a | a | e | c | b |
| b | b | c | e | a |
| c | c | b | a | e |
Group Structure and Isomorphisms
The Klein four-group is one of the two groups of order 4 up to isomorphism, the other being the cyclic group \mathbb{Z}_4.[3] This classification follows from the fact that any group of order 4 is either cyclic or consists of elements all of order at most 2, leading uniquely to the non-cyclic abelian structure of the Klein four-group.[5] The Klein four-group is isomorphic to the direct product \mathbb{Z}_2 \times \mathbb{Z}_2 of two cyclic groups of order 2.[7] In additive notation, the elements of \mathbb{Z}_2 \times \mathbb{Z}_2 are pairs (m,n) where m,n \in \{0,1\}, and the group operation is defined by (m,n) + (p,q) = (m + p \pmod{2}, n + q \pmod{2}). [8] This isomorphism highlights the Klein four-group's structure as an elementary abelian 2-group, where every non-identity element has order 2.[9] The subgroup structure of the Klein four-group consists of three proper nontrivial subgroups, each of order 2 and generated by one of the non-identity elements.[10] These subgroups are all isomorphic to \mathbb{Z}_2 and intersect trivially at the identity, forming a partition of the non-identity elements.[11] All quotient groups of the Klein four-group by its nontrivial proper subgroups are isomorphic to \mathbb{Z}_2.[3] Since the group is abelian, every subgroup is normal, and the quotient by any order-2 subgroup has order 2, yielding the unique group structure of that order.[7]Elementary Properties
The Klein four-group is abelian, so its center is the entire group.[7] All non-identity elements have order 2, meaning that the square of any such element is the identity.[7] The derived subgroup, generated by all commutators, is trivial because the group is abelian.[7] The automorphism group of the Klein four-group is isomorphic to the symmetric group S_3 on three letters and thus has order 6; this arises from the action permuting the three non-identity elements while preserving the group operation.[12] Any homomorphism from the Klein four-group to another group has a kernel that is either trivial, one of its order-2 subgroups, or the whole group, yielding an image isomorphic to the Klein four-group (injective case), the cyclic group of order 2, or the trivial group, respectively. All such images are abelian groups of exponent 2.[12]Presentations
Abstract Presentation
The Klein four-group admits the abstract presentation \langle a, b \mid a^2 = b^2 = (ab)^2 = [e](/page/Identity_element) \rangle, where e is the identity element. This presentation defines the group as the quotient of the free group on generators a and b by the normal closure of the specified relations. The relations impose that a and b each have order 2, and their product ab also has order 2. The relation (ab)^2 = [e](/page/Identity_element) implies that the group is abelian, since it yields abab = [e](/page/Identity_element), and conjugating by a (using a^2 = [e](/page/Identity_element)) gives bab = a, so ab = ba.[13] Equivalently, in additive notation, the Klein four-group is the free abelian group of rank 2 modulo the subgroup generated by twice each basis element, i.e., \mathbb{Z}^2 / 2\mathbb{Z}^2. To verify this presentation using von Dyck's theorem, note that any group generated by elements satisfying these relations admits a homomorphism from the presented group onto it; since the concrete Klein four-group has order 4 and satisfies the relations, the presented group is isomorphic to it. The Klein four-group arises as a quotient of the dihedral group of order 8 by its center, which has order 2.[14] It also appears as a normal subgroup of the dihedral group of order 8.[15]Concrete Realizations
The Klein four-group admits several explicit constructions as familiar algebraic structures. One standard realization is as the direct product \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}, consisting of the set \{(0,0), (1,0), (0,1), (1,1)\} equipped with componentwise addition modulo 2.[2] Here, the identity is (0,0), and each non-identity element has order 2, since adding any such element to itself yields (0,0). The group operation is abelian, reflecting the direct product structure. The addition table for this realization is as follows:| + | (0,0) | (1,0) | (0,1) | (1,1) |
|---|---|---|---|---|
| (0,0) | (0,0) | (1,0) | (0,1) | (1,1) |
| (1,0) | (1,0) | (0,0) | (1,1) | (0,1) |
| (0,1) | (0,1) | (1,1) | (0,0) | (1,0) |
| (1,1) | (1,1) | (0,1) | (1,0) | (0,0) |
| · | e | a | b | c |
|---|---|---|---|---|
| e | e | a | b | c |
| a | a | e | c | b |
| b | b | c | e | a |
| c | c | b | a | e |
Representations
Permutation Representations
The Klein four-group admits a faithful permutation representation of degree 4, realized as the normal subgroup V = \{\id, (1\,2)(3\,4), (1\,3)(2\,4), (1\,4)(2\,3)\} of the symmetric group S_4.[12] This embedding consists of the identity and the three double transpositions in S_4, each of which is an even permutation as the product of two disjoint transpositions.[12] The subgroup V is unique up to conjugacy in S_4 and serves as the canonical faithful representation of the Klein four-group in the symmetric group on four letters.[3] The action of V on the set \{1,2,3,4\} via these permutations is faithful and transitive, yielding a single orbit of size 4.[12] No non-identity element of V fixes any point in the set, so the stabilizer of any point is trivial. By the orbit-stabilizer theorem, the orbit size equals the group order divided by the stabilizer order, confirming | \Orb(x) | = |V| / |\Stab_V(x)| = 4 / 1 = 4 for any x \in \{1,2,3,4\}.[12] This faithful representation arises from the Cayley embedding of the Klein four-group into S_4, where the group acts regularly on itself by left multiplication, producing a homomorphism with trivial kernel.[12] Since V is normal in S_4, the latter acts on V by conjugation, preserving the subgroup structure. For \sigma \in S_4 and g \in V, the conjugated element is given by \sigma \cdot g = \sigma g \sigma^{-1} \in V. This action transitively permutes the three non-identity elements of V, as all double transpositions are conjugate in S_4.[12][3]Linear Representations
The Klein four-group, being abelian, admits only one-dimensional irreducible representations over the complex numbers \mathbb{C}. There are exactly four such irreducible representations, each corresponding to a group homomorphism from the group to the multiplicative group \mathbb{C}^\times, with images in the roots of unity of order dividing 2, i.e., \{\pm 1\}. These are the trivial representation \rho_1, where every element acts as multiplication by 1, and three non-trivial representations: \rho_2, where one generator acts as 1 and the other as -1 (with their product acting as -1); \rho_3, where the generators act as -1 and 1, respectively (product -1); and \rho_4, where both generators act as -1 (product 1).[17][18] The character table of the Klein four-group over \mathbb{C} encodes these representations and is given below, with rows corresponding to the irreducible characters and columns to the conjugacy classes (the identity and the three elements of order 2):| Character | e | a | b | ab |
|---|---|---|---|---|
| \chi_1 (trivial) | 1 | 1 | 1 | 1 |
| \chi_2 | 1 | 1 | -1 | -1 |
| \chi_3 | 1 | -1 | 1 | -1 |
| \chi_4 | 1 | -1 | -1 | 1 |