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Group isomorphism

In abstract algebra, a group isomorphism is a bijective homomorphism between two groups, meaning it is a one-to-one and onto function that preserves the group operation by mapping the product of any two elements in the first group to the product of their images in the second group.^[1] Groups related by an isomorphism are denoted as isomorphic (often written G \cong H) and possess identical algebraic structures, differing only in the notation or labeling of their elements.^[2] Isomorphisms form an equivalence relation on the class of all groups, being reflexive (every group is isomorphic to itself via the identity map), symmetric (if G \cong H, then H \cong G via the inverse map), and transitive (if G \cong H and H \cong K, then G \cong K via the composition of maps).^[1] They preserve fundamental properties, including the order of the group, the orders of individual elements, the identity element, inverses, abelianness, cyclicity, and the existence and orders of subgroups.^[2] ^[3] This preservation ensures that isomorphic groups are indistinguishable in terms of their abstract behavior and structural features.^[4] A central goal in group theory is the classification of groups up to isomorphism, which seeks to identify and enumerate all distinct group structures by considering isomorphic groups as equivalent.^[5] Notable examples include the isomorphism between the additive group of real numbers (\mathbb{R}, +) and the multiplicative group of positive real numbers (\mathbb{R}^+, \times) via the exponential function f(x) = e^x, demonstrating how different operations can underlie the same structure.^[2] For small finite orders, such as 1, 2, or 3, there is a unique group up to isomorphism in each case, highlighting the finiteness of isomorphism classes for low orders.^[1]

Core Definitions

Group Homomorphisms

A group homomorphism is a function \phi: G \to H between two groups G and H that preserves the group operation, meaning \phi(ab) = \phi(a)\phi(b) for all a, b \in G./11:_Homomorphisms/11.01:_Group_Homomorphisms) This condition implies that \phi maps the identity element e_G of G to the identity element e_H of H, since setting a = b = e_G yields \phi(e_G) = \phi(e_G)^2, so \phi(e_G) must be e_H.^[6] Homomorphisms preserve additional structural features of groups. Specifically, if an element a \in G has finite order n, then the order of \phi(a) divides n, because if a^n = e_G, applying \phi gives \phi(a)^n = e_H./11:_Homomorphisms/11.01:_Group_Homomorphisms) Moreover, the image of any subgroup K \leq G under \phi, denoted \phi(K) = \{\phi(k) \mid k \in K\}, forms a subgroup of H.^[6] The kernel of a homomorphism \phi: G \to H, denoted \ker \phi = \{g \in G \mid \phi(g) = e_H\}, is the set of elements in G that map to the identity in H. This kernel is always a normal subgroup of G, as for any g \in G and k \in \ker \phi, one has \phi(gkg^{-1}) = \phi(g)\phi(k)\phi(g)^{-1} = \phi(g)e_H\phi(g)^{-1} = e_H, so gkg^{-1} \in \ker \phi./11:_Homomorphisms/11.01:_Group_Homomorphisms) The image of \phi, denoted \operatorname{im} \phi = \{\phi(g) \mid g \in G\}, is a subgroup of H.^[6] For finite groups, the order of G satisfies |G| = |\ker \phi| \cdot |\operatorname{im} \phi|, which follows from the fact that the cosets of \ker \phi in G are in bijection with the elements of \operatorname{im} \phi, applying Lagrange's theorem to the index of the kernel. Isomorphisms are precisely the bijective group homomorphisms./11:_Homomorphisms/11.01:_Group_Homomorphisms)

Group Isomorphisms

A group isomorphism is a bijective group homomorphism \phi: G \to H between two groups G and H.^[7]^[1] While homomorphisms preserve the group operation in one direction, isomorphisms ensure a two-way structural correspondence due to their bijectivity.^[7] For such a \phi to qualify as an isomorphism, its inverse \phi^{-1}: H \to G must also be a homomorphism.^[7] To see this, let a', b' \in H; set a = \phi^{-1}(a') and b = \phi^{-1}(b'). Then \phi^{-1}(a'b') = \phi^{-1}(\phi(a)\phi(b)) = \phi^{-1}(\phi(ab)) = ab = \phi^{-1}(a')\phi^{-1}(b'), confirming that \phi^{-1} preserves the operation.^[7] Additionally, since \phi maps the identity e_G to e_H, the inverse maps e_H back to e_G.^[7] If an isomorphism exists between G and H, the groups are denoted G \cong H and are said to be structurally equivalent, meaning they share the same abstract group structure despite potentially different underlying sets.^[7]^[8] This equivalence implies that G and H cannot be distinguished by any intrinsic group-theoretic properties. Isomorphisms preserve all fundamental group-theoretic properties.^[1] Specifically, they preserve the order of the group |G| = |H| due to bijectivity, and the order of elements |g| = |\phi(g)| for all g \in G, as powers and the identity are mapped accordingly.^[1] Subgroups of G correspond bijectively to subgroups of H via \phi, since the image of a subgroup under a homomorphism is a subgroup, and bijectivity ensures a one-to-one matching.^[1] Similarly, cosets are preserved: the cosets of a subgroup K \leq G map to the cosets of \phi(K) \leq H.^[1] Relations between elements, such as commutators [g, h] = g^{-1}h^{-1}gh mapping to [\phi(g), \phi(h)], are also maintained, ensuring that properties like abelianness hold equivalently in both groups.^[7]^[1]

Fundamental Properties

Preservation of Structure

An isomorphism \phi: G \to H between groups G and H preserves the group operation pointwise, meaning \phi(ab) = \phi(a)\phi(b) for all a, b \in G.^[9] This property directly implies that \phi maps the identity element of G to the identity of H and inverses to inverses, ensuring the structural integrity of the operation is maintained across the groups.^[10] Extending this, powers are preserved such that \phi(a^n) = \phi(a)^n for any integer n, as the operation's associativity and the homomorphism property allow iterative application.^[9] Consequently, the order of an element is invariant: if a \in G has finite order m, then \phi(a) has order m in H.^[11] Beyond elements, isomorphisms preserve the subgroup lattice bijectively: if K is a subgroup of G, then \phi(K) is a subgroup of H, and the inverse map \phi^{-1} ensures the correspondence is one-to-one.^[9] This bijectivity stems from the isomorphism's invertibility, mapping subgroups to subgroups without loss or addition of structure.^[10] Moreover, normal subgroups are preserved: if K \trianglelefteq G, then \phi(K) \trianglelefteq H, as the normality condition—conjugation by group elements—translates under the structure-preserving map.^[10] Several key invariants remain unchanged under isomorphism. The center Z(G) = \{ z \in G \mid zg = gz \ \forall g \in G \} of G maps to the center Z(H) of H, since it is a characteristic subgroup fully determined by the group's operation.^[9] Similarly, the derived subgroup G' = \langle [g,h] \mid g,h \in G \rangle, generated by commutators, is preserved as \phi(G') = H', reflecting its role as the smallest normal subgroup capturing non-abelian structure.^[9] The abelianization G/G', the largest abelian quotient of G, is thus isomorphic to H/H'.^[10] The exponent of the group, defined as the least common multiple of the orders of its elements (or infinite if unbounded), is also invariant, following from the preservation of individual element orders.^[9] For cyclic subgroups, an isomorphism satisfies \langle \phi(a) \rangle = \phi(\langle a \rangle) for any a \in G, with generators mapping to generators: if a generates \langle a \rangle, then \phi(a) generates \phi(\langle a \rangle).^[10] This ensures that the cyclic structure, including relations among powers, is fully transferred.^[9]

Isomorphisms as Equivalence

In group theory, the relation of isomorphism defines an equivalence relation on the class of all groups. Specifically, two groups G and H are related if there exists a group isomorphism \phi: G \to H. This relation is reflexive, symmetric, and transitive.^[12] To see that the relation is reflexive, consider the identity map \mathrm{id}_G: G \to G defined by \mathrm{id}_G(g) = g for all g \in G. This map is a bijection, and it preserves the group operation since \mathrm{id}_G(gg') = gg' = \mathrm{id}_G(g) \mathrm{id}_G(g') for all g, g' \in G. Thus, \mathrm{id}_G is an isomorphism, so G \cong G.^[13] The relation is symmetric: if \phi: G \to H is an isomorphism, then \phi^{-1}: H \to G exists as a bijection. Moreover, \phi^{-1} preserves the operation because \phi^{-1}(h h') = \phi^{-1}(h) \phi^{-1}(h') follows from applying \phi to both sides and using the homomorphism property of \phi. Hence, H \cong G.^[12] Transitivity holds as follows: if \phi: G \to H and \psi: H \to K are isomorphisms, then the composition \psi \circ \phi: G \to K is a bijection (since both \phi and \psi are bijections). It preserves the operation because

(\psi \circ \phi)(g g') = \psi(\phi(g g')) = \psi(\phi(g) \phi(g')) = \psi(\phi(g)) \psi(\phi(g')) = (\psi \circ \phi)(g) (\psi \circ \phi)(g')

for all g, g' \in G. Thus, G \cong K.^[13] This equivalence relation partitions the class of all groups into equivalence classes, known as isomorphism classes. The isomorphism class of a group G, denoted [G], consists of all groups H such that H \cong G. Groups within the same isomorphism class are structurally identical, meaning they share all group-theoretic properties preserved by isomorphisms, such as order, the existence of certain subgroups, or solvability.^[12] The concept of isomorphism classes has profound implications for the classification of groups. Classification efforts aim to describe groups up to isomorphism using invariants, such as the group's order, whether it is abelian or non-abelian, or its presentation by generators and relations. Groups in the same class are considered indistinguishable, allowing mathematicians to focus on representatives of each class rather than enumerating all possible groups. This approach underpins major results, like the complete classification of finite simple groups.

Illustrative Examples

Basic Examples

A fundamental example of a group isomorphism involves the additive group of integers and its subgroups. Consider the map \phi: (\mathbb{Z}, +) \to (n\mathbb{Z}, +) defined by \phi(x) = n x, where n is a fixed non-zero integer. This map is a homomorphism because \phi(x + y) = n (x + y) = n x + n y = \phi(x) + \phi(y). It is injective, as the kernel is trivial: if \phi(x) = 0, then n x = 0, so x = 0 since \mathbb{Z} has no zero divisors. It is also surjective, as every element n k in n\mathbb{Z} is \phi(k) for k \in \mathbb{Z}. Thus, \phi establishes that (\mathbb{Z}, +) \cong (n\mathbb{Z}, +).^[14] Another basic example is the isomorphism between the Klein four-group V_4 = \{e, a, b, ab\}, where a^2 = b^2 = e and ab = ba, and the direct product \mathbb{Z}_2 \times \mathbb{Z}_2 = \{(0,0), (1,0), (0,1), (1,1)\} with componentwise addition modulo 2. The explicit isomorphism is given by \phi(e) = (0,0), \phi(a) = (1,0), \phi(b) = (0,1), and \phi(ab) = (1,1). To verify it is a homomorphism, note that, for instance, \phi(a \cdot a) = \phi(e) = (0,0) and \phi(a) + \phi(a) = (1,0) + (1,0) = (0,0); similarly for other products like a \cdot b = ab mapping to (1,0) + (0,1) = (1,1). The map is bijective, as it pairs each element uniquely and covers all four elements of the codomain, with trivial kernel confirming injectivity.^[7] For finite cyclic groups, the additive group \mathbb{Z}_4 = \{0, 1, 2, 3\} with addition modulo 4 is isomorphic to itself via the identity map \phi(k) = k, which trivially preserves the operation, is injective and surjective, and has order 4 elements like 1 with order 4. In contrast, \mathbb{Z}_4 is not isomorphic to \mathbb{Z}_2 \times \mathbb{Z}_2, as the latter has all non-identity elements of order 2 (e.g., (1,0) + (1,0) = (0,0)), while \mathbb{Z}_4 has an element of order 4, violating preservation of element orders under any potential isomorphism.^[7]

Non-Trivial Examples

One prominent non-trivial example of a group isomorphism is between the symmetric group S_3, consisting of all permutations of three elements, and the dihedral group D_3 of order 6, which describes the symmetries of an equilateral triangle. The isomorphism maps the permutations in S_3 to compositions of rotations and reflections in D_3; specifically, the identity maps to the identity, the 3-cycles (123) and (132) map to rotations by 120° and 240° respectively, and the transpositions (12), (13), (23) map to reflections across the corresponding axes.^[15] This bijection preserves the group operation, as verified by checking that the relations in both presentations hold equivalently.^[16] In the infinite case, the free group F_2 on two generators embeds as a subgroup of the special linear group SL(2, \mathbb{Z}), which consists of $2 \times 2 integer matrices with determinant 1. A concrete realization is the Sanov subgroup generated by the matrices \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} and \begin{pmatrix} 1 & 0 \\ 2 & 1 \end{pmatrix}, which is free of rank 2, establishing the isomorphism F_2 \cong \langle A, B \rangle \leq SL(2, \mathbb{Z}).^[17] Another striking infinite example involves the additive group of real numbers (\mathbb{R}, +), which is isomorphic to a proper subgroup of itself. Such an isomorphism can be constructed non-constructively using Hamel bases over \mathbb{Q}: both \mathbb{R} and the proper subgroup have Hamel bases of cardinality of the continuum; extend a bijection between the bases linearly to obtain a \mathbb{Q}-linear isomorphism. However, this relies on the axiom of choice to guarantee the existence of such bases.^[17] A non-abelian example highlighting structural complexity is the embedding of the special orthogonal group SO(3), the group of 3D rotations preserving orientation, as a subgroup of the general linear group GL(3, \mathbb{R}). Explicitly, elements of SO(3) are represented by rotation matrices, such as the rotation by angle \theta around the z-axis given by \begin{pmatrix} \cos \theta & -\sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{pmatrix}, which satisfy A^T A = I and \det A = 1, confirming the subgroup property under matrix multiplication.^[18] These examples appear non-trivial because the groups often admit vastly different presentations or geometric interpretations, yet they share identical multiplication tables (for finite cases) or structural properties up to relabeling of elements, underscoring that isomorphism equates abstract structures despite concrete dissimilarities.^[15]

Isomorphism Theorems

First Isomorphism Theorem

The First Isomorphism Theorem, also known as the Fundamental Homomorphism Theorem, establishes a deep connection between group homomorphisms, their kernels, and quotient groups. It states that if \phi: G \to H is a group homomorphism, then the kernel K = \ker \phi is a normal subgroup of G, and there is an isomorphism \overline{\phi}: G/K \to \operatorname{im} \phi given by \overline{\phi}(gK) = \phi(g) for all g \in G.^[19] To verify that \overline{\phi} is well-defined, suppose g_1 K = g_2 K for some g_1, g_2 \in [G](/page/G). Then g_1^{-1} g_2 \in K, so \phi(g_1^{-1} g_2) = [e_H](/page/Identity), where e_H is the identity in H. Thus, \phi(g_1) = \phi(g_2), implying \overline{\phi}(g_1 K) = \overline{\phi}(g_2 K).^[19] Next, \overline{\phi} is a homomorphism because it preserves the group operation: for cosets g_1 K, g_2 K \in G/K,

\overline{\phi}(g_1 K \cdot g_2 K) = \overline{\phi}(g_1 g_2 K) = \phi(g_1 g_2) = \phi(g_1) \phi(g_2) = \overline{\phi}(g_1 K) \cdot \overline{\phi}(g_2 K).

This confirms that \overline{\phi} respects the multiplication in the quotient group and the image subgroup.^[19] The map \overline{\phi} is injective: if \overline{\phi}(g_1 K) = \overline{\phi}(g_2 K), then \phi(g_1) = \phi(g_2), so \phi(g_1^{-1} g_2) = e_H and g_1^{-1} g_2 \in K, hence g_1 K = g_2 K. It is surjective onto \operatorname{im} \phi because for any h \in \operatorname{im} \phi, there exists g \in G such that \phi(g) = h, so \overline{\phi}(g K) = h. Therefore, \overline{\phi} is an isomorphism.^[19] A key corollary is that every homomorphic image of a group G is isomorphic to some quotient group of G. Specifically, \operatorname{im} \phi \cong G / \ker \phi, which provides a powerful method for classifying groups up to isomorphism by identifying them with familiar quotient structures.^[19]

Second and Third Isomorphism Theorems

The second isomorphism theorem provides a relationship between a subgroup and a normal subgroup of a group, describing an isomorphism between certain quotient groups. Let G be a group, H \leq G a subgroup, and N \trianglelefteq G a normal subgroup. Then HN = \{hn \mid h \in H, n \in N\} is a subgroup of G, N \trianglelefteq HN, and H \cap N \trianglelefteq H. Moreover, there is a group isomorphism

HN / N \cong H / (H \cap N)

given by the map \phi: H \to HN / N defined by \phi(h) = hN, which is well-defined because N \trianglelefteq G.^[20]^[21] To see that \phi is a homomorphism, note that for h_1, h_2 \in H, \phi(h_1 h_2) = h_1 h_2 N = (h_1 N)(h_2 N) = \phi(h_1) \phi(h_2), since coset multiplication is preserved. The map \phi is surjective because every element of HN / N is of the form h'n' for some h' \in H, n' \in N, which equals h'N = \phi(h'). The kernel of \phi is \ker \phi = \{h \in H \mid hN = N\} = H \cap N. By the first isomorphism theorem applied to \phi, the image is isomorphic to H / \ker \phi = H / (H \cap N), and since the image is all of HN / N, the desired isomorphism holds. This establishes the coset equality and normality preservation in the product subgroup.^[20]^[21] The third isomorphism theorem extends this by relating quotients in a chain of normal subgroups, showing compatibility under successive factoring. Let G be a group with normal subgroups N \trianglelefteq K \trianglelefteq G. Then K / N \trianglelefteq G / N, and there is a group isomorphism

(G / N) / (K / N) \cong G / K

induced by the natural projection \pi: G \to G / K given by \pi(g) = gK, which factors through G / N. Specifically, the isomorphism \psi: (G / N) / (K / N) \to G / K is defined by \psi((gN)(K / N)) = gK for g \in G.^[20]^[21] The map \psi arises from the composition of the natural surjection G / N \to G / K (with kernel K / N) and the quotient map G \to G / N. To verify, first define \overline{\pi}: G / N \to G / K by \overline{\pi}(gN) = gK; this is a well-defined homomorphism because if gN = g'N, then g^{-1}g' \in N \subseteq K, so gK = g'K. It is surjective since every gK equals \overline{\pi}(gN). The kernel is \{\, gN \mid gK = K \,\} = K / N. Applying the first isomorphism theorem to \overline{\pi} yields (G / N) / \ker \overline{\pi} \cong G / K, which is the stated isomorphism. This confirms the normality of K / N in G / N via coset containment: for g \in G, n \in N, k \in K, (gN)(kN)(gN)^{-1} = (gkg^{-1})N \in K / N since K \trianglelefteq G and N \subseteq K.^[20]^[21]

Special Cases

Cyclic Groups

A cyclic group is generated by a single element, and all finite cyclic groups of the same order are isomorphic to each other and to the additive group \mathbb{Z}_n. Specifically, if G = \langle g \rangle where g has order n, then there exists an isomorphism \phi: \mathbb{Z}_n \to G defined by \phi(k) = g^k for k = 0, 1, \dots, n-1. This mapping preserves the group operation because \phi(k + m \mod n) = g^{k+m} = g^k g^m = \phi(k) \phi(m), and it is bijective since g generates G and has order n.^[22] Similarly, all infinite cyclic groups are isomorphic to the additive group of integers \mathbb{Z}. For an infinite cyclic group G = \langle a \rangle with a \neq e, the map \phi: \mathbb{Z} \to G given by \phi(k) = a^k is an isomorphism, as it is a bijective homomorphism: surjective because every element is a power of a, injective because the kernel is trivial (no nonzero k satisfies a^k = e), and it preserves addition via the group operation.^[22] The classification of cyclic groups up to isomorphism is thus determined solely by their order: there is exactly one isomorphism class for each finite order n (namely, \mathbb{Z}_n) and one for the infinite case (\mathbb{Z}). No two cyclic groups of the same order are non-isomorphic, as the explicit isomorphisms above establish equivalence. This follows from the first isomorphism theorem applied to the natural surjection from \mathbb{Z} (or \mathbb{Z}_n) onto the cyclic group, yielding the kernel as the appropriate multiple of the order.^[23] The first isomorphism theorem also applies directly to quotients of cyclic groups. For d \mid n, the subgroup \langle d \rangle of \mathbb{Z}_n has order n/d, and the quotient \mathbb{Z}_n / \langle d \rangle \cong \mathbb{Z}_{d}, as the natural projection map from \mathbb{Z}_n to \mathbb{Z}_{d} (reduction modulo d) has kernel \langle d \rangle. This isomorphism highlights how cyclic groups' structure simplifies under quotients by cyclic subgroups.^[24] To contrast, non-cyclic groups of the same order as a cyclic one are not isomorphic to it. For example, the Klein four-group \mathbb{Z}_2 \times \mathbb{Z}_2 has order 4 but is not cyclic, as all its non-identity elements have order 2, whereas \mathbb{Z}_4 has an element of order 4; thus, they are not isomorphic since isomorphisms preserve element orders. Specifically, \mathbb{Z}_2 \times \mathbb{Z}_2 has three elements of order 2, while \mathbb{Z}_4 has only one.^[25]

Automorphisms

An automorphism of a group G is a bijective homomorphism \alpha: G \to G, preserving the group operation in both directions.^[26] The collection of all such automorphisms, denoted \Aut(G), forms a group under function composition, where the binary operation is defined by (\alpha \circ \beta)(x) = \alpha(\beta(x)) for all x \in G, the identity element is the identity map \id_G, and the inverse of \alpha is \alpha^{-1}, which exists since \alpha is bijective.^[27] Automorphisms represent the symmetries of the group structure itself, capturing ways to relabel elements while maintaining the operation.^[26] A key subgroup of \Aut(G) consists of the inner automorphisms, generated by conjugation: for each g \in G, the map c_g: x \mapsto g x g^{-1} is an automorphism, and the set \Inn(G) = \{ c_g \mid g \in G \} forms a normal subgroup of \Aut(G).^[27] The conjugation action defines a homomorphism from G to \Aut(G) via g \mapsto c_g, with kernel equal to the center Z(G) = \{ z \in G \mid z x = x z \ \forall x \in G \}; by the first isomorphism theorem, this yields \Inn(G) \cong G / Z(G).^[28] Automorphisms outside \Inn(G), called outer automorphisms, exist when \Aut(G) / \Inn(G) is nontrivial, measuring "external" symmetries beyond conjugation.^[28] For the cyclic group \mathbb{Z}_n under addition modulo n, the automorphisms are multiplication by integers k coprime to n: \phi_k(a) = k a \mod n, and \Aut(\mathbb{Z}_n) \cong U(n), the multiplicative group of units modulo n.^[29] Here, the order of \Aut(\mathbb{Z}_n) is \phi(n), Euler's totient function, illustrating how automorphisms quantify the invertible scalings preserving the cyclic structure.^[29] The holomorph of G, denoted \Hol(G) = G \rtimes \Aut(G), is the semidirect product where \Aut(G) acts on G by evaluation, providing a unified framework to study both the group elements and their symmetries as a single larger group.^[30]

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[29]
holomorph of a group - PlanetMath.org
Mar 22, 2013 · If G G is a group, the holomorph of G G is the group G⋊θAut(G) G ⋊ θ Aut ⁡ ( G ) under the identity map θ:Aut(G)→Aut(G) θ : Aut ⁡ ( G ) ...