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Automorphism group

In mathematics, particularly in abstract algebra and category theory, the automorphism group of a mathematical object—such as a group, ring, vector space, or graph—is the set of all isomorphisms from the object to itself, forming a group under the operation of function composition.^[1] These automorphisms capture the intrinsic symmetries of the structure, preserving its operations and relations while potentially rearranging elements.^[2] For a group G, the automorphism group \operatorname{Aut}(G) consists precisely of the group isomorphisms \phi: G \to G, and it plays a central role in understanding the group's structure and classifications.^[1] A key distinction within \operatorname{Aut}(G) is between inner automorphisms, which are conjugations by elements of G (i.e., \phi_g(h) = ghg^{-1} for fixed g \in G), forming the inner automorphism group \operatorname{Inn}(G) isomorphic to G/Z(G) where Z(G) is the center of G, and outer automorphisms, which are the coset representatives in the quotient \operatorname{Out}(G) = \operatorname{Aut}(G)/\operatorname{Inn}(G).^[2] Inner automorphisms always exist and reflect the group's own action on itself, while outer ones, if nontrivial, reveal additional symmetries beyond conjugation.^[1] Notable examples illustrate the diversity of automorphism groups. For the infinite cyclic group \mathbb{Z}, \operatorname{Aut}(\mathbb{Z}) \cong C_2, generated by the inversion map n \mapsto -n.^[2] For finite cyclic groups \mathbb{Z}_n, \operatorname{Aut}(\mathbb{Z}_n) is isomorphic to the multiplicative group of units U(n) modulo n, with order \phi(n) where \phi is Euler's totient function; for instance, \operatorname{Aut}(\mathbb{Z}_5) \cong C_4.^[2] In contrast, nonabelian groups like the dihedral group D_3 (symmetries of an equilateral triangle) have \operatorname{Aut}(D_3) \cong D_3 itself, with six elements.^[2] These structures highlight how \operatorname{Aut}(G) encodes information about generators and relations in G, often determined by where automorphisms send generating sets.^[2] Beyond groups, automorphism groups extend to other algebraic objects; for example, in Lie algebras, \operatorname{Aut}(L) comprises Lie algebra isomorphisms from L to itself, with applications in representation theory and geometry.^[3] The study of automorphism groups is fundamental in classification problems, such as determining when two groups are isomorphic or computing rigidity in geometric contexts, and it intersects with broader areas like Galois theory, where \operatorname{Aut}(K/F) for field extensions describes symmetries of roots.^[1]

Fundamentals

Definition

In mathematics, an isomorphism between two mathematical objects is a bijective mapping that preserves the structure of the objects, such as their operations or relations.^[4] An automorphism of a mathematical object X is an isomorphism from X to itself, representing a symmetry of the object that leaves its essential properties unchanged.^[5] The automorphism group of X, denoted \Aut(X) (or sometimes \Gamma(X)), is the set of all automorphisms of X equipped with the group operation of function composition.^[1] This forms a group because composition of automorphisms is again an automorphism, ensuring closure; function composition is associative; the identity mapping on X serves as the identity element; and every automorphism, being a bijective isomorphism, has an inverse that is also an automorphism.^[2]

Basic properties

The automorphism group \operatorname{Aut}(X) of an algebraic structure X (such as a group, ring, or vector space) with underlying set X embeds naturally as a subgroup of the symmetric group \operatorname{Sym}(X) on X. This embedding arises from the action of \operatorname{Aut}(X) on X by evaluation: for \phi \in \operatorname{Aut}(X) and x \in X, define \phi \cdot x = \phi(x). Since every automorphism is a bijection preserving the structure of X, this induces a permutation of the elements of X, and the resulting homomorphism \operatorname{Aut}(X) \to \operatorname{Sym}(X) is injective, making \operatorname{Aut}(X) isomorphic to its image, a subgroup of \operatorname{Sym}(X).^[6] The center Z(\operatorname{Aut}(X)) consists of those automorphisms in \operatorname{Aut}(X) that commute with every element of \operatorname{Aut}(X) under composition. An element \phi \in Z(\operatorname{Aut}(X)) satisfies \phi \circ \psi = \psi \circ \phi for all \psi \in \operatorname{Aut}(X), meaning \phi centralizes the entire group action. In many cases, particularly when the center of the underlying structure X is trivial, Z(\operatorname{Aut}(X)) is also trivial, reflecting a lack of non-trivial automorphisms that act compatibly with all symmetries.^[7] If the underlying set X is finite, then \operatorname{Aut}(X) is finite, with order |\operatorname{Aut}(X)| \leq |X|!, and more precisely, |\operatorname{Aut}(X)| divides |X|! as a consequence of being a subgroup of the finite group \operatorname{Sym}(X). For structures with infinite underlying sets, \operatorname{Aut}(X) is typically infinite, though exceptions exist (e.g., the automorphism group of the integers under addition, which has order 2).^[6] The natural action of \operatorname{Aut}(X) on X by automorphisms is faithful: the kernel of this action, consisting of automorphisms that fix every element of X pointwise, is trivial. This follows directly from the injectivity of the embedding into \operatorname{Sym}(X), as any structure-preserving bijection fixing all elements must be the identity map.^[6]

Examples

Automorphisms of finite groups

The automorphism group of the cyclic group \mathbb{Z}/n\mathbb{Z} is isomorphic to the multiplicative group of units modulo n, denoted (\mathbb{Z}/n\mathbb{Z})^*, which consists of integers coprime to n under multiplication modulo n and has order given by Euler's totient function \phi(n). This isomorphism arises because automorphisms correspond to multiplication by units, preserving the group structure. For instance, when n = p is prime, \operatorname{Aut}(\mathbb{Z}/p\mathbb{Z}) \cong \mathbb{Z}/(p-1)\mathbb{Z}, a cyclic group of order p-1. For non-abelian finite groups, the symmetric group S_n provides key examples. The automorphism group \operatorname{Aut}(S_n) is isomorphic to S_n itself for n \neq 6, reflecting the highly symmetric nature of permutations where inner automorphisms capture all symmetries. However, S_6 is exceptional: \operatorname{Aut}(S_6) has order $2|S_6|, twice that of S_6, due to an outer automorphism arising from the transitive action on 6-element sets distinct from the standard permutation representation. The alternating group A_n, the subgroup of even permutations, also exhibits rigid automorphism structures for larger n. Specifically, \operatorname{Aut}(A_n) \cong S_n for n \geq 7, with the outer automorphisms arising from conjugation by elements of S_n \setminus A_n.^[8] Finite abelian groups offer further illustrations, particularly through their primary decomposition. For a finite abelian p-group decomposed into cyclic factors via invariant factors, say G \cong \mathbb{Z}/p^{a_1}\mathbb{Z} \times \cdots \times \mathbb{Z}/p^{a_k}\mathbb{Z} with a_1 \geq \cdots \geq a_k \geq 1, the automorphism group \operatorname{Aut}(G) can be computed as a matrix group over \mathbb{Z}/p\mathbb{Z} acting on the factors, preserving the exponents. A concrete case is the Klein four-group V_4 \cong \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}, whose automorphism group is isomorphic to S_3, the symmetric group on 3 letters, of order 6; this reflects the action of permuting the three non-identity elements.

Automorphisms of graphs and geometric objects

In graph theory, an automorphism of a simple undirected graph G = (V, E) is a bijective mapping \phi: V \to V such that for any distinct vertices u, v \in V, the pair \{u, v\} is an edge in E if and only if \{\phi(u), \phi(v)\} is an edge in E.^[9] The set of all such automorphisms forms the automorphism group \operatorname{Aut}(G), which acts faithfully as a permutation group on the vertex set V.^[9] The order of this group, |\operatorname{Aut}(G)|, quantifies the graph's symmetries and relates to the size of its isomorphism class through the orbit-stabilizer theorem, where the number of distinct labelings of isomorphic graphs is n! / |\operatorname{Aut}(G)| for n = |V|.^[10] Representative examples illustrate these symmetries. The complete graph K_n on n vertices has \operatorname{Aut}(K_n) \cong S_n, the symmetric group on n elements, since any permutation of vertices preserves all possible edges.^[11] In contrast, the cycle graph C_n for n \geq 3 has \operatorname{Aut}(C_n) \cong D_{2n}, the dihedral group of order $2n, generated by rotations and reflections that preserve the cyclic structure.^[12] Automorphism groups also describe symmetries of geometric objects modeled as graphs. For polyhedra, the automorphism group includes transformations like rotations and reflections that preserve the edge structure. The cube, with 8 vertices and 12 edges, has full symmetry group \operatorname{Aut}(\text{[cube](/page/Cube)}) \cong S_4 \times \mathbb{Z}/2\mathbb{Z} of order 48, where S_4 accounts for rotational symmetries permuting the 4 space diagonals, and the \mathbb{Z}/2\mathbb{Z} factor incorporates reflections.^[13] A key result connecting group theory and graph symmetries is Frucht's theorem, which asserts that every finite group is isomorphic to \operatorname{Aut}(G) for some finite simple undirected graph G.^[11] This theorem, proved in 1939, highlights the expressive power of graph automorphisms in realizing arbitrary finite group structures.^[11] To compute \operatorname{Aut}(G), one approach uses the adjacency matrix A of G, an n \times n symmetric matrix with A_{ij} = 1 if \{i, j\} \in E and 0 otherwise. A permutation \pi of the vertices induces a permutation matrix P such that \pi is an automorphism if and only if [P A P^T = A](/page/If_and_only_if), meaning P conjugates A to itself.^[14] This matrix formulation facilitates algorithmic enumeration of automorphisms by testing permutations that preserve the matrix structure.^[14]

Inner and outer automorphisms

Inner automorphisms

In group theory, an inner automorphism of a group G is an automorphism \phi: G \to G of the form \phi(g) = h^{-1} g h for all g \in G, where h is a fixed element of G.^[15] This conjugation action defines a map from G to the automorphism group \Aut(G), and the image of this map forms the subgroup \Inn(G) consisting of all inner automorphisms. The concept extends to other algebraic structures, such as rings or Lie algebras, where conjugation by an invertible element is similarly defined and yields an automorphism.^[15] The subgroup \Inn(G) is isomorphic to the quotient group G / Z(G), where Z(G) denotes the center of G, the set of elements that commute with every element of G. To see this, consider the map \phi: G \to \Inn(G) defined by \phi(a) = i_a, where i_a(g) = a g a^{-1} for all g \in G. This \phi is a group homomorphism because i_{ab}(g) = ab g (ab)^{-1} = a (b g b^{-1}) a^{-1} = i_a \circ i_b(g). The kernel of \phi is precisely Z(G), since i_a = i_e (the identity automorphism) if and only if a g a^{-1} = g for all g \in G, meaning a \in Z(G). Moreover, \phi is surjective onto \Inn(G) by construction. By the first isomorphism theorem, G / \ker(\phi) \cong \im(\phi), so G / Z(G) \cong \Inn(G).^[16] The subgroup \Inn(G) is always normal in \Aut(G), as for any \pi \in \Aut(G) and inner automorphism \phi_x(g) = x g x^{-1} with x \in G, the conjugate \pi \circ \phi_x \circ \pi^{-1} = \phi_{\pi(x)} is again inner.^[17] A concrete example occurs with the symmetric group S_3 of order 6, which has trivial center Z(S_3) = \{e\} since no non-identity element commutes with all permutations. Thus, \Inn(S_3) \cong S_3 / Z(S_3) \cong S_3, implying \Inn(S_3) has order 6. In fact, every automorphism of S_3 is inner, as the natural conjugation map S_3 \to \Aut(S_3) is an isomorphism, confirmed by noting that automorphisms permute the three transpositions (which generate S_3) in a way that matches the action of S_3 itself.^[18] The inner automorphisms arise naturally from the adjoint representation of G, which embeds G into \Aut(G) via the conjugation action: the map \Ad: G \to \Aut(G) sends g \mapsto \Ad_g, where \Ad_g(h) = g h g^{-1}. The image of \Ad is precisely \Inn(G), providing a linear perspective in the case of Lie groups, where the differential of \Ad yields the adjoint action on the Lie algebra.^[19]

Outer automorphisms

The outer automorphism group of a group G, denoted \Out(G), is defined as the quotient group \Aut(G) / \Inn(G), where \Aut(G) is the full automorphism group and \Inn(G) is the normal subgroup of inner automorphisms.^[20] Elements of \Out(G) are thus cosets of \Inn(G) in \Aut(G), representing equivalence classes of automorphisms where two automorphisms are equivalent if one is obtained from the other by composition with an inner automorphism (i.e., conjugation by an element of G). This structure captures the "non-internal" symmetries of G, distinguishing automorphisms that cannot be realized by conjugation within the group itself.^[20] The group \Out(G) quantifies symmetries beyond those induced by the group's own elements, and it is often trivial, meaning every automorphism of G is inner. For instance, in many finite groups, including most symmetric groups, all automorphisms arise from conjugations. However, non-trivial outer automorphisms exist in specific cases, highlighting exceptional symmetries. For abelian groups, where the center Z(G) = G and thus \Inn(G) is trivial, \Out(G) \cong \Aut(G), so outer automorphisms coincide with all automorphisms. A representative example is the Klein four-group V_4 \cong \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}, for which \Aut(V_4) \cong S_3 (the symmetric group on three letters), yielding \Out(V_4) \cong S_3. This isomorphism reflects the action of automorphisms permuting the three non-identity elements of order 2.^[21] A classic non-abelian example occurs with the symmetric group S_6, where \Out(S_6) \cong \mathbb{Z}/2\mathbb{Z}, while \Out(S_n) is trivial for all n \neq 6. This exceptional outer automorphism interchanges the conjugacy classes of transpositions (order 2 elements of cycle type (2)) and double transpositions (products of three disjoint transpositions, also order 2 but of cycle type (2,2,2)), which have the same size in S_6 unlike in other S_n. Constructions of this automorphism include actions on sets of pentads (subsets of five elements) or via embeddings into larger permutation groups, confirming its uniqueness up to composition with inner automorphisms.^[22] For finite simple groups, outer automorphism groups are typically small, often trivial, reflecting their rigid structure. By the classification of finite simple groups, \Out(G) = 1 for most such G, but non-trivial cases arise, such as \Out(\PSL(2,7)) \cong \mathbb{Z}/2\mathbb{Z}, generated by a field automorphism or duality in the projective special linear group. Among sporadic simple groups, 14 of the 26 have trivial \Out(G), while the remaining 12 have \Out(G) \cong \mathbb{Z}/2\mathbb{Z}, as in the Mathieu group M_{12} or the Harada-Norton group. These non-trivial outer automorphisms often stem from graph automorphisms or field extensions in the underlying Lie type structures.^[23]^[24]

Applications in algebra

Automorphism groups of rings and fields

In ring theory, an automorphism of a ring R is a bijective ring homomorphism from R to itself, which preserves both the addition and multiplication operations.^[25] For commutative rings that are algebras over a base field, such automorphisms typically fix the elements of the base field, as they must map the multiplicative identity to itself and preserve scalar multiplication.^[25] The automorphism group \operatorname{Aut}(K) of a field K consists of all field automorphisms of K, which are bijective maps preserving addition, multiplication, and the multiplicative identity.^[26] For finite fields \mathbb{F}_{p^n}, where p is prime and n \geq 1, this group is cyclic of order n and isomorphic to \mathbb{Z}/n\mathbb{Z}.^[27] It is generated by the Frobenius automorphism \phi: \mathbb{F}_{p^n} \to \mathbb{F}_{p^n} defined by \phi(x) = x^p, which satisfies \phi(ab) = \phi(a)\phi(b) and \phi(a + b) = \phi(a) + \phi(b) due to the freshman's dream identity (a + b)^p = a^p + b^p in characteristic p.^[27] More generally, for \mathbb{F}_q with q = p^n, \operatorname{Aut}(\mathbb{F}_q) = \{\phi^k \mid 0 \leq k < n\}, where \phi^k(x) = x^{p^k}. For infinite fields, the automorphism groups can be trivial or vastly larger depending on the field and any imposed conditions. The group \operatorname{Aut}(\mathbb{Q}) of the rational numbers is trivial, consisting only of the identity map, since any automorphism fixes 1 and hence all integers by induction, and thus all rationals of the form a/b.^[26] Similarly, \operatorname{Aut}(\mathbb{R}) of the real numbers is trivial without assuming continuity: automorphisms fix \mathbb{Q}, preserve positivity (as squares map to squares), and thus fix all reals by density of rationals.^[26] In contrast, \operatorname{Aut}(\mathbb{C}) of the complex numbers, without continuity assumptions, is enormous, with cardinality exceeding that of the continuum, arising from the transcendence degree of \mathbb{C} over \mathbb{Q}, which allows arbitrary permutations of transcendence bases.^[28] In Galois theory, the automorphism group \operatorname{Aut}(K/F) of a field extension K/F—fixing the base field F pointwise—coincides with the Galois group when the extension is Galois (normal and separable).^[29] The absolute case, where F is the prime subfield of K, recovers \operatorname{Aut}(K) as the full group of automorphisms.^[29]

Automorphism groups of vector spaces

In a vector space V over a field F, an automorphism is an invertible linear map T: V \to V that preserves the vector space structure, and the set of all such automorphisms forms the automorphism group \operatorname{Aut}(V) under composition.^[30] This group is isomorphic to the general linear group \operatorname{GL}(V), consisting of all invertible linear operators on V.^[30] For a finite-dimensional vector space V of dimension n over F, \operatorname{Aut}(V) \cong \operatorname{GL}(n, F), the group of n \times n invertible matrices over F.^[30] When F is the finite field \mathbb{F}_q of order q, the order of this group is |\operatorname{GL}(n, q)| = \prod_{k=0}^{n-1} (q^n - q^k), which counts the number of ordered bases for V.^[31] In the infinite-dimensional case, \operatorname{Aut}(V) comprises all invertible linear operators on V, but studies often restrict to those preserving a chosen Hamel basis or satisfying additional conditions like continuity in topological settings.^[32] The choice of a basis for V identifies \operatorname{Aut}(V) with \operatorname{GL}(n, F) in the finite-dimensional case, where automorphisms act as matrix multiplications relative to that basis; changing the basis corresponds to conjugation by the change-of-basis matrix.^[30] For the standard space F^n, elements of \operatorname{Aut}(F^n) act by permuting coordinates via matrix multiplication.^[31] Special subgroups include the orthogonal group \operatorname{O}(n, F), which stabilizes a non-degenerate symmetric bilinear form, and the symplectic group \operatorname{Sp}(n, F), which stabilizes a non-degenerate alternating bilinear form.^[30]

Category-theoretic aspects

Automorphisms in categories

In category theory, an automorphism of an object X in a category \mathcal{C} is defined as an isomorphism f: X \to X, which is equivalently an endomorphism that admits an inverse morphism within \mathcal{C}.^[33] The collection of all such automorphisms, denoted \mathrm{Aut}_\mathcal{C}(X), forms a group under the composition of morphisms, with the identity morphism serving as the neutral element and inverses provided by the isomorphism property; this group is a subgroup of the monoid \mathrm{Hom}_\mathcal{C}(X, X) consisting of all endomorphisms.^[33] Automorphisms exhibit naturality in the sense that they are compatible with the morphisms of the category. This compatibility ensures that automorphisms act as symmetries internal to the categorical framework.^[33] Illustrative examples appear in concrete categories: in the category \mathbf{Set} of sets, \mathrm{Aut}(X) consists of all bijections from X to itself, yielding the symmetric group \mathrm{Sym}(X); in the category \mathbf{Grp} of groups, it comprises group isomorphisms from a group to itself; and in the category \mathbf{Top} of topological spaces, automorphisms are homeomorphisms of the space.^[33] These instances highlight how \mathrm{Aut}_\mathcal{C}(X) captures the invertible symmetries specific to each category's notion of isomorphism. In general, \mathrm{Aut}_\mathcal{C}(X) functions as an automorphism group in the sense of ordinary 0-categories, where composition yields the group operation.^[33] Regarding monoidal structures, when \mathcal{C} is monoidal, automorphisms may interact with the tensor product, for instance preserving it up to natural isomorphism in contexts like sheaf toposes, where they relate to automorphism sheaves.^[33] Extending to 2-categories, the notion of automorphism generalizes to auto-equivalences, which are equivalences F: \mathcal{C} \to \mathcal{C} (functors invertible up to natural isomorphism), contrasting with strict automorphisms that are strictly invertible functors without needing weak inverses.^[34] Strictification theorems allow many 2-categories to be equivalent to strict ones, wherein strict automorphism 2-groups emerge, comprising strict equivalences and invertible 2-morphisms under horizontal composition.^[34] For example, in the strict 2-category of groups \mathbf{Grp}_2, the automorphism 2-group of a group H corresponds to the crossed module (H \to \mathrm{Aut}(H)).^[34]

Automorphism group functor

In category theory, the automorphism group functor Aut is defined as a contravariant functor from a category \mathcal{C} to the category of groups \mathbf{Grp}, typically restricted to the wide subcategory of \mathcal{C} consisting of all objects and only the isomorphisms as morphisms (known as the core of \mathcal{C}, or equivalently via the opposite category \mathcal{C}^{\mathrm{op}} since inverting all isomorphisms yields an equivalent structure). For an object X \in \mathrm{Ob}(\mathcal{C}), \mathrm{Aut}(X) is the group of all isomorphisms X \to X under composition. For an isomorphism f: X \to Y in \mathcal{C}, the induced group homomorphism \mathrm{Aut}(f): \mathrm{Aut}(Y) \to \mathrm{Aut}(X) is given by conjugation: \mathrm{Aut}(f)(g) = f^{-1} \circ g \circ f for g \in \mathrm{Aut}(Y). This assignment preserves composition and identities, making Aut functorial on this subcategory, though it does not extend to arbitrary morphisms in general categories like \mathbf{Grp}.^[35]^[36] This contravariant nature arises because a morphism f: X \to Y in \mathcal{C} "pulls back" automorphisms from Y to X, reversing the direction; viewing it covariantly on \mathcal{C}^{\mathrm{op}} aligns the maps with the reversed arrows. In algebraic categories, such as those of modules or varieties, Aut often admits additional structure, being representable as a functor (isomorphic to a Hom-functor into a group object), which provides a universal property for automorphism groups via the forgetful functor to the underlying category. For instance, the endomorphism functor \mathrm{End}(-) is representable, and Aut extracts the group of units therein.^[35] A concrete example occurs in the category \mathbf{Grp} of groups. The functor Aut sends a group G to its automorphism group \mathrm{Aut}(G), the group of group isomorphisms G \to G. For an isomorphism \phi: G \to H, \mathrm{Aut}(\phi): \mathrm{Aut}(H) \to \mathrm{Aut}(G) is \beta \mapsto \phi^{-1} \circ \beta \circ \phi for \beta \in \mathrm{Aut}(H), preserving the group operation. This induced map reflects how isomorphisms between groups conjugate their respective automorphism groups, highlighting the functor's role in preserving algebraic structure across isomorphic objects.^[35]

References

[1]
Group Automorphism -- from Wolfram MathWorld
A group automorphism is an isomorphism from a group to itself. If G is a finite multiplicative group, an automorphism of G can be described as a way of ...
[2]
[PDF] Lecture 4.6: Automorphisms - Mathematical and Statistical Sciences
Definition. An automorphism is an isomorphism from a group to itself. The set of all automorphisms of G forms a group, called the automorphism group of. G ...
[3]
Automorphism Group - an overview | ScienceDirect Topics
The automorphism group of a Lie algebra $ L $ is defined as the group of all isomorphisms from $ L $ onto itself, which forms a group under the ...
[4]
Isomorphism -- from Wolfram MathWorld
Formally, an isomorphism is bijective morphism. Informally, an isomorphism is a map that preserves sets and relations among elements. " A is isomorphic to B " ...
[5]
Automorphism -- from Wolfram MathWorld
An automorphism is an isomorphism of a system of objects onto itself. The term derives from the Greek prefix alphaupsilontauomicron (auto) "self"
[6]
[PDF] Abstract Algebra
a divides b the greatest common divisor of a, b also the ideal generated by a, b the order of the set A, the order of the element x.
[7]
ON THE CENTRE OF THE AUTOMORPHISM GROUP OF A GROUP
We study the structure of the centre of the automorphism group of a group G when the centre of G is a cyclic group.
[8]
[PDF] Automorphism Group of Graphs (Supplemental Material for Intro to ...
Jan 15, 2018 · A graph automorphism is simply an isomorphism from a graph to itself. In other words, an automorphism on a graph G is a bijection φ : V (G) ...
[9]
[PDF] on the group-theoretic properties of the automorphism groups of ...
The orbit-stabilizer theorem provides an extremely simple way to compute the order of any graph's automorphism group. We shall make use of it in the next.
[10]
[PDF] automorphism groups of simple graphs
In the field of Abstract Algebra, one of the fundamental concepts is that of combining particular sets with operations on their elements and studying the ...
[11]
[PDF] CS E6204 Lecture 5 Automorphisms
Aut(Cn). ∼. = Dn. 5. Page 6. Example The automorphism group of the n-cycle Cn is the dihedral group Dn with 2n elements. Figure 4: Aut(C5) = D5. Example [Fr37] ...
[12]
[PDF] Matroid Automorphisms and Symmetry Groups - Webbox
Cube or octahedron: symmetry group. ∼. = Z3. 2 o S3. ∼. = S4 × Z2;. Dodecahedron ... We also remark that the matroid automorphism groups Aut(MW ) for the ...
[13]
[PDF] Math 443/543 Graph Theory Notes 5: Graphs as matrices, spectral ...
Nov 3, 2014 · Similarly, the permutation matrix gives an isomorphism. Now we see that the adjacency matrix can be used to count uv-walks.<|control11|><|separator|>
[14]
Inner Automorphism -- from Wolfram MathWorld
An inner automorphism of a group G is an automorphism of the form phi(g)=h^(-1)gh, where h is a fixed element of G.
[15]
G/Z(G) is Isomorphic to Inn(G) - Mathonline
Proof: From proposition 1 we know that $G/Z(G)$ is isomorphic to $\mathrm{Inn}(G)$. So if $\mathrm{Inn}(G)$ is abelian then $G/Z(G)$ is abelian. But ...Missing: theory | Show results with:theory
[16]
inner automorphism - PlanetMath
Mar 22, 2013 · An automorphism that isn't inner is called an outer automorphism. The composition operation gives the set of all automorphisms of G G ...
[17]
[PDF] The Automorphism Tower Problem Simon Thomas
(a) Aut(G) is a complete group. (b) Inn(G) is a characteristic subgroup of Aut(G). Proof. If Aut(G) is a complete group, then the normal subgroup Inn(G) is.
[18]
[PDF] Algebra Final Exam Solutions 1. Automorphisms of groups. (a) Define
(b) Prove or disprove that every automorphism of S3 is inner. Solution: This is true. For every group G we have a natural map α:G → Aut(G) sending h to ...
[19]
The Adjoint Representation of a Lie Group - SpringerLink
Every group G acts on itself by inner automorphisms: the map associated with an element g is h ↦ ghg −1. If G is a Lie group, the differential of each inner ...
[20]
Outer Automorphism Group -- from Wolfram MathWorld
The outer automorphism group is the quotient group Aut(G)/Inn(G), which is the automorphism group of G modulo its inner automorphism group.Missing: definition | Show results with:definition
[21]
[PDF] Groups - LSU Math
(2) The group Z2 × Z2 is isomorphic to the Klein 4-group. Therefore, the two nonisomorphic groups of order 4 are Z4 and Z2 × Z2. (3) All the hypotheses in ...
[22]
[PDF] Exercises on the outer automorphism of S6
In this class, we investigate the outer automorphism of S6. Let's recall some definitions, so that we can state what an outer automorphism is. Definition 1.1.
[23]
Linear group L2(7) = L3(2) - ATLAS
Sep 14, 2004 · PGL2(7) are preimages C and D where D has order 3. Automorphisms. An outer automorphism, u say, of L2(7) = L3(2) of order 2 may be obtained ...
[24]
None
### Summary of Outer Automorphism Groups of Sporadic Simple Groups
[25]
[PDF] Chapter 1 - Rings
Let R and S be commutative rings. A ring homomorphism from R to S is a function h: R → S that is operation preserving for both addition and multiplication. That ...
[26]
[PDF] FIELD AUTOMORPHISMS OF R AND Qp - Keith Conrad
Theorem 2.1. The only field homomorphism Q → Q is the identity. In particular, the only field automorphism of Q is the identity. Proof.
[27]
[PDF] Finite Fields
Definition: Frobenius Automorphism. Let F be a field of characteristic p. The Frobenius automorphism of F is the function ϕ: F → F defined by. ϕ(a) = ap.
[28]
[PDF] AUTOMORPHISMS OF THE COMPLEX NUMBERS
On the other hand, if we consider arbitrary automor- phisms of C with no continuity requirement, then the automorphism group of C is huge. One illustration ...
[29]
[PDF] THE GALOIS CORRESPONDENCE 1. Introduction Let L/K be a field ...
When L/K is a Galois extension, the group Aut(L/K) is denoted Gal(L/K) and is called the Galois group of the extension. For a finite Galois extension L/K ...<|control11|><|separator|>
[30]
[PDF] Endomorphisms, Automorphisms, and Change of Basis
The set of all endomorphisms of V will be denoted by L (V;V ). A vector space isomorphism that maps V to itself is called an automorphism of V .Missing: abstract | Show results with:abstract
[31]
[PDF] LECTURE II 1. General Linear Group Let Fq be a finite field of order ...
Let Fq be a finite field of order q. Then GLn(q), the general linear group over the field. Fq, is the group of invertible n × n matrices with coefficients in Fq ...
[32]
[PDF] Infinite dimensional linear groups: how we have studied them
Let V be a vector space over a field F. The general linear group GL(V,F) of V over V consists of all F–automorphisms of V made group under composition of maps ...
[33]
automorphism in nLab
Oct 5, 2025 · 1. Definition. An automorphism of an object x in a category C is an isomorphism f : x → x f : x \to x . In other words, an automorphism is an ...
[34]
automorphism 2-group in nLab
### Summary of Automorphisms in 2-Categories
[35]
[PDF] Category Theory in Context Emily Riehl
Mar 1, 2014 · ... contravariant functor F from C to D is a functor F : Cop → D.26 ... automorphism group Aut(G), the group of isomorphisms ϕ: G → G in ...
[36]
[PDF] What is Category Theory?
Although Eilenberg and Mac Lane introduced functor categories, they did not ... Finally, the automorphism group A(G) of a group G is a functor only if one ...