Fact-checked by Grok 2 weeks ago

Inner automorphism

In group theory, an inner automorphism of a group G is an automorphism \phi_g \in \operatorname{Aut}(G) defined by conjugation with a fixed element g \in G, specifically \phi_g(h) = g h g^{-1} for all h \in G.^[1] The set of all such inner automorphisms forms the inner automorphism group \operatorname{Inn}(G), which is a subgroup of the full automorphism group \operatorname{Aut}(G).^[2] The inner automorphism group \operatorname{Inn}(G) is isomorphic to the quotient group G / Z(G), where Z(G) denotes the center of G, consisting of all elements that commute with every element of G.^[2] This isomorphism arises from the homomorphism \psi: G \to \operatorname{Aut}(G) given by \psi(g) = \phi_g, whose kernel is precisely Z(G) and whose image is \operatorname{Inn}(G).^[3] Consequently, \operatorname{Inn}(G) is trivial (i.e., consists only of the identity automorphism) if and only if G is abelian.^[4] \operatorname{Inn}(G) is always a normal subgroup of \operatorname{Aut}(G), and the quotient group \operatorname{Out}(G) = \operatorname{Aut}(G) / \operatorname{Inn}(G) is called the outer automorphism group, which captures the automorphisms of G that cannot be realized by conjugation.^[2] Inner automorphisms play a fundamental role in classifying group structures, as they reveal symmetries inherent to the group's own elements and help distinguish abelian from non-abelian groups, while outer automorphisms highlight additional, "exotic" symmetries beyond conjugation.^[5]

Definition and Basics

Formal Definition

An automorphism of a group G is an isomorphism from G to itself, that is, a bijective group homomorphism \phi: G \to G satisfying \phi(xy) = \phi(x)\phi(y) for all x, y \in G.^[6] An inner automorphism of a group G is a group automorphism \phi_g: G \to G defined by conjugation with a fixed element g \in G, given by \phi_g(x) = gxg^{-1} for all x \in G.^[7] This map is a homomorphism because

\phi_g(xy) = g(xy)g^{-1} = (gxg^{-1})(gyg^{-1}) = \phi_g(x)\phi_g(y)

for all x, y \in G, and it is bijective with inverse \phi_{g^{-1}}, since \phi_g \circ \phi_{g^{-1}} = \mathrm{id}_G = \phi_{g^{-1}} \circ \phi_g.^[7]^[8] The set of all inner automorphisms of G, denoted \mathrm{Inn}(G), forms a subgroup of the automorphism group \mathrm{Aut}(G) under composition.^[7] It contains the identity automorphism \phi_e = \mathrm{id}_G, where e is the identity element of G; it is closed under composition because \phi_g \circ \phi_h = \phi_{gh} for all g, h \in G; and it is closed under inverses because the inverse of \phi_g is \phi_{g^{-1}}.^[9]

Initial Examples

To illustrate inner automorphisms, consider the conjugation map in a group G, defined by \phi_g(h) = g h g^{-1} for g, h \in G. The assignment g \mapsto \phi_g defines a group homomorphism from G to \Aut(G), the automorphism group of G, whose image consists of all inner automorphisms and whose kernel is precisely the center Z(G) of G.^[10] A concrete example arises in the symmetric group S_3, which consists of all permutations of three elements and has order 6. Conjugation by the transposition (1\ 2) sends the transposition (1\ 3) to (2\ 3), since (1\ 2)(1\ 3)(1\ 2)^{-1} = (2\ 3). This relabeling of elements demonstrates a non-trivial inner automorphism, as it permutes the three transpositions in S_3 while preserving the group structure.^[11] In contrast, the Klein four-group V_4 = \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}, with elements \{e, a, b, c\} where each non-identity element has order 2 and the product of any two distinct non-identity elements is the third, is abelian. Thus, its center is the entire group Z(V_4) = V_4, making the kernel of the conjugation homomorphism equal to V_4 and all inner automorphisms trivial.^[12] The quaternion group Q_8 = \{\pm 1, \pm i, \pm j, \pm k\} with relations i^2 = j^2 = k^2 = -1, ij = k, jk = i, and ki = j provides another example where inner automorphisms are non-trivial. Here, the center is Z(Q_8) = \{\pm 1\}, and the inner automorphism group \Inn(Q_8) is isomorphic to V_4, capturing the action of conjugation by elements outside the center on the non-central elements.^[13]

Automorphism Groups

Inner Automorphism Group

The inner automorphism group \operatorname{[Inn](/page/Inn)}(G) of a group G is isomorphic to the quotient group G/Z(G), where Z(G) = \{ z \in G \mid zg = gz \text{ for all } g \in G \} denotes the center of G.^[14] This isomorphism follows from the conjugation homomorphism \phi: G \to \operatorname{[Aut](/page/Automorphism)}(G) defined by \phi(g)(h) = ghg^{-1} for all g, h \in G. The image of \phi is precisely \operatorname{[Inn](/page/Inn)}(G), and the kernel is Z(G). By the first isomorphism theorem for groups, G / \ker(\phi) \cong \operatorname{im}(\phi), so G/Z(G) \cong \operatorname{[Inn](/page/Inn)}(G).^[14] \operatorname{Inn}(G) forms a normal subgroup of the full automorphism group \operatorname{Aut}(G). To see this, consider any \alpha \in \operatorname{Aut}(G) and inner automorphism \phi_g \in \operatorname{Inn}(G) given by conjugation by g \in G. Then \alpha \circ \phi_g \circ \alpha^{-1} = \phi_{\alpha(g)}, which is again an inner automorphism. Thus, \operatorname{Inn}(G) is invariant under conjugation by elements of \operatorname{Aut}(G). The quotient group \operatorname{Aut}(G)/\operatorname{Inn}(G) is called the outer automorphism group and denoted \operatorname{Out}(G).^[15] Since \operatorname{Inn}(G) \cong G/Z(G), it inherits key properties from this quotient: for instance, \operatorname{Inn}(G) is abelian if and only if G/Z(G) is abelian. For a finite group G, the order satisfies |\operatorname{Inn}(G)| = |G| / |Z(G)|.^[14]

Outer Automorphism Group

The outer automorphism group of a group G, denoted \operatorname{Out}(G), is defined as the quotient group \operatorname{Aut}(G) / \operatorname{Inn}(G), where \operatorname{Aut}(G) denotes the full automorphism group of G and \operatorname{Inn}(G) is the normal subgroup consisting of all inner automorphisms.^[16] This construction identifies automorphisms that differ only by composition with an inner automorphism, so elements of \operatorname{Out}(G) are cosets \phi \operatorname{Inn}(G) for \phi \in \operatorname{Aut}(G), representing equivalence classes of automorphisms up to conjugation by elements of G.^[16] The group operation on these cosets is induced by composition of automorphisms, making \operatorname{Out}(G) a group that captures the "outer" symmetries of G. A concrete example illustrates this definition for the symmetric group S_3, which has order 6 and consists of all permutations of three elements. The automorphism group \operatorname{Aut}(S_3) is isomorphic to S_3 itself, as any automorphism must permute the three transpositions (the generators of order 2) while preserving the group relations.^[17] Since S_3 has trivial center, \operatorname{Inn}(S_3) \cong S_3 / Z(S_3) \cong S_3, and thus the quotient \operatorname{Out}(S_3) is the trivial group.^[17] This computation shows that all automorphisms of S_3 arise from inner ones, reflecting the complete symmetry captured by conjugation within the group. The significance of \operatorname{Out}(G) lies in its role as a measure of symmetries beyond those induced by the group's own elements via conjugation; a non-trivial \operatorname{Out}(G) signals additional structural features, such as embeddings into larger groups or unexpected isomorphisms, that reveal deeper properties of G.^[18] For instance, in the classification of finite simple groups, the solvability of outer automorphism groups provides key constraints on possible group structures.^[18] Moreover, \operatorname{Out}(G) acts naturally on the set of conjugacy classes of G, since inner automorphisms preserve these classes and any automorphism maps conjugacy classes to conjugacy classes of the same size; this action is well-defined on the quotient and often permutes classes in ways that inner automorphisms cannot.^[19]

Structural Relations

Connection to Center

The center Z(G) of a group G, consisting of all elements that commute with every element of G, serves as the kernel of the natural conjugation homomorphism \alpha: G \to \Aut(G) defined by \alpha(g)(h) = g h g^{-1} for all g, h \in G.^[20] This map embeds G into its automorphism group via inner automorphisms, with elements of Z(G) inducing the identity automorphism.^[3] If Z(G) = G, then G is abelian, and the conjugation map is trivial, implying that the inner automorphism group \Inn(G) is also trivial.^[21] Conversely, if Z(G) = \{e\}, the group is centerless, and the conjugation map yields an isomorphism G \cong \Inn(G).^[22] In general, \Inn(G) \cong G / Z(G).^[23] Inner automorphisms preserve the center setwise and, in fact, fix it pointwise: for any g \in G and z \in Z(G), the inner automorphism \phi_g(z) = g z g^{-1} = z, since z commutes with g.^[3] A concrete illustration occurs in extraspecial p-groups, which are non-abelian p-groups of order p^{2m+1} with center Z(G) cyclic of order p; here, \Inn(G) \cong G / Z(G) has order p^{2m}.^[21]^[22]

Relation to Conjugacy Classes

Inner automorphisms act on the group G by evaluation, meaning that for \phi \in \operatorname{Inn}(G) and g \in G, the action is \phi \cdot g = \phi(g).^[24] The conjugacy class of an element x \in G is precisely the orbit of x under this action, consisting of all elements \phi(x) for \phi \in \operatorname{Inn}(G).^[12] This partitions G into conjugacy classes, each corresponding to the equivalence relation where two elements are conjugate if one is the image of the other under some inner automorphism.^[24] The size of the conjugacy class of x, denoted \operatorname{cl}(x), is given by the index of the centralizer C_G(x) = \{ g \in G \mid gx = xg \} in G:

|\operatorname{cl}(x)| = [G : C_G(x)].

^[12] This formula arises because C_G(x) is the stabilizer of x under the conjugation action, and by the orbit-stabilizer theorem, the orbit size equals the index of the stabilizer.^[24] A subgroup N \leq G is normal if and only if it is preserved setwise by every inner automorphism, meaning \phi(N) = N for all \phi \in \operatorname{Inn}(G).^[24] Equivalently, N is a union of conjugacy classes, ensuring invariance under conjugation by elements of G.^[12] For finite groups, the number of conjugacy classes equals the number of irreducible complex representations, a consequence of Burnside's theorem in representation theory.^[25]

Special Cases in Groups

Finite p-Groups

In finite p-groups of order greater than 1, the center Z(G) is non-trivial, ensuring that the inner automorphism group \operatorname{Inn}(G) \cong G/Z(G) is a proper quotient of G. For non-abelian finite p-groups, the derived subgroup G' is likewise non-trivial. Moreover, inner automorphisms act trivially on the abelianization G/G', since G/G' is abelian. These features distinguish inner automorphisms from the broader automorphism group, often leading to non-trivial outer automorphisms that act non-trivially on Z(G) or G'.^[26] A fundamental structural theorem is due to Gaschütz, which asserts that every non-abelian finite p-group admits outer automorphisms of p-power order; more broadly, for any finite p-group G not isomorphic to the cyclic group of order p, the order of the outer automorphism group \operatorname{Out}(G) is divisible by p. This guarantees the existence of non-inner automorphisms whenever |G| > p, highlighting cases where \operatorname{Out}(G) is non-trivial and contributes additional p-power structure to \operatorname{Aut}(G). For non-abelian examples, |\operatorname{Aut}(G)| is divisible by p (in fact, by higher powers), as is |\operatorname{Inn}(G)|, with the outer component providing the extra factors.^[23] Representative examples illustrate when \operatorname{Out}(G) \cong \mathbb{Z}_p. The dihedral group D_4 of order 8 (with p=2) has \operatorname{Aut}(D_4) \cong D_4 of order 8 and \operatorname{Inn}(D_4) \cong \mathbb{Z}_2 \times \mathbb{Z}_2 of order 4, yielding \operatorname{Out}(D_4) \cong \mathbb{Z}_2. This outer automorphism corresponds to an inversion that swaps the two conjugacy classes of reflections while fixing rotations, demonstrating a minimal non-trivial outer action in a small non-abelian 2-group.^[23]

Non-Abelian Simple Groups

Non-abelian simple groups possess no nontrivial normal subgroups and are non-commutative, which forces their center Z(G) to be trivial. As a result, the conjugation action yields an isomorphism G \cong \operatorname{Inn}(G), since the kernel of the map G \to \operatorname{Aut}(G) given by g \mapsto c_g (where c_g(h) = ghg^{-1}) is precisely Z(G) = \{e\}. The full automorphism group \operatorname{Aut}(G) fits into the short exact sequence $1 \to \operatorname{Inn}(G) \to \operatorname{Aut}(G) \to \operatorname{Out}(G) \to 1, where \operatorname{Out}(G) denotes the outer automorphism group. For many non-abelian simple groups, this sequence splits, yielding \operatorname{Aut}(G) \cong \operatorname{Inn}(G) \rtimes \operatorname{Out}(G). A prominent example is the alternating group A_5, the smallest non-abelian simple group of order 60, whose outer automorphism group is \operatorname{Out}(A_5) \cong \mathbb{Z}_2. This nontrivial outer automorphism is exceptional, stemming from the embedding A_5 \trianglelefteq S_5 and interchanging two conjugacy classes of elements of order 5.^[27] In stark contrast, the Monster group M, the largest of the 26 sporadic simple groups with order approximately $8 \times 10^{53}, has trivial outer automorphism group, so \operatorname{Aut}(M) = \operatorname{Inn}(M) \cong M. This completeness property underscores the Monster's role as a "rigid" structure in the classification.^[28] The classification of finite simple groups reveals that outer automorphisms of sporadic groups frequently connect to symmetries of underlying geometric or combinatorial objects, such as graphs; for instance, in the fourth Fischer group \mathrm{Fi}_{24}', graph automorphisms influence the structure of centralizers within its automorphism group.^[29]

Generalizations

Lie Algebras

In the context of Lie algebras, the notion of inner automorphisms from group theory generalizes to inner derivations, which arise from the Lie bracket in a manner analogous to conjugation by group elements. For a Lie algebra \mathcal{L} over a field k of characteristic zero, an inner derivation is defined via the adjoint map \mathrm{ad}_x: \mathcal{L} \to \mathcal{L} given by \mathrm{ad}_x(y) = [x, y] for all x, y \in \mathcal{L}, where [ \cdot, \cdot ] denotes the Lie bracket. This map is a derivation because it satisfies the Leibniz rule \mathrm{ad}_x([y, z]) = [\mathrm{ad}_x(y), z] + [y, \mathrm{ad}_x(z)], which follows directly from the Jacobi identity. The collection \mathrm{Inn}(\mathcal{L}) = \{ \mathrm{ad}_x \mid x \in \mathcal{L} \} forms a Lie subalgebra of the full derivation algebra \mathrm{Der}(\mathcal{L}), consisting of all k-linear endomorphisms D: \mathcal{L} \to \mathcal{L} that preserve the bracket via D([y, z]) = [D(y), z] + [y, D(z)].^[30] The structure of \mathrm{Inn}(\mathcal{L}) is closely tied to the center of \mathcal{L}, defined as

Z(\mathcal{L}) = \{ z \in \mathcal{L} \mid [z, y] = 0 \ \forall \, y \in \mathcal{L} \}.

The adjoint representation \mathrm{ad}: \mathcal{L} \to \mathrm{gl}(\mathcal{L}) has kernel precisely Z(\mathcal{L}), and its image is \mathrm{Inn}(\mathcal{L}) as a Lie subalgebra of \mathrm{Der}(\mathcal{L}). Thus, there is a Lie algebra isomorphism

\mathrm{Inn}(\mathcal{L}) \cong \mathcal{L} / Z(\mathcal{L}),

reflecting how central elements act trivially via the adjoint action. This quotient captures the "effective" inner derivations modulo the center.^[31] A concrete example is the Lie algebra \mathfrak{sl}(2, \mathbb{R}) of $2 \times 2 real matrices with trace zero, equipped with the commutator bracket. This algebra has trivial center Z(\mathfrak{sl}(2, \mathbb{R})) = \{ 0 \}, as any element commuting with the standard basis \{ h = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, x = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, y = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix} \} must be scalar, but trace zero forces it to be zero. Consequently, the adjoint map is injective, yielding \mathrm{Inn}(\mathfrak{sl}(2, \mathbb{R})) \cong \mathfrak{sl}(2, \mathbb{R}).^[31]^[32] For semisimple Lie algebras, inner derivations exhaust all derivations: \mathrm{Der}(\mathcal{L}) = \mathrm{Inn}(\mathcal{L}). This follows from the nondegeneracy of the Killing form and the absence of nonzero abelian ideals, implying no outer derivations exist and the outer derivation algebra is trivial. Semisimple Lie algebras, direct sums of simple ones, thus have rigid derivation structures determined entirely by their own elements.^[31]

Other Algebraic Structures

In ring theory, an inner automorphism of an associative unital ring R is defined as conjugation by a unit u \in R^\times, given by \phi_u(r) = u r u^{-1} for all r \in R. The set of all such maps forms the inner automorphism group \operatorname{Inn}(R), which is isomorphic to the quotient R^\times / Z(R^\times), where Z(R^\times) denotes the center of the unit group. These automorphisms fix the center Z(R) pointwise and play a key role in understanding the structure of the full automorphism group \operatorname{Aut}(R).^[33] A representative example occurs in the matrix ring M_n(k) over a field k. Here, the inner automorphisms are precisely the conjugations by invertible matrices, and \operatorname{Inn}(M_n(k)) is isomorphic to the projective general linear group \operatorname{PGL}_n(k) = \operatorname{GL}_n(k)/k^\times. In fact, all k-algebra automorphisms of M_n(k) are inner.^[34] In the context of modules, the notion of inner automorphisms extends analogously through the endomorphism ring \operatorname{End}_R(M) of an R-module M. Inner automorphisms of \operatorname{End}_R(M) are induced by conjugation by automorphisms of M, reflecting the symmetries of the module structure.^[35] For categories, particularly groupoids, inner automorphisms are captured by natural isomorphisms arising from conjugation functors. Specifically, an inner automorphism of a groupoid \mathcal{G} is induced by conjugation by an object or morphism, yielding a natural isomorphism between the identity functor and a conjugation functor, generalizing the group case categorically.^[36] For division rings, the outer automorphism group \operatorname{Out}(R) = \operatorname{Aut}(R)/\operatorname{Inn}(R) can be non-trivial; for instance, in quaternion algebras over number fields with non-trivial Galois groups, field automorphisms of the center may extend to the algebra, producing outer automorphisms.^[37]

References

[1]
[PDF] Automorphism groups - Missouri State University
The automorphism group of G is written Aut(G). ▷ The inner automorphism group of G, written Inn(G), is the group of automorphisms of the form fg (x) = gxg−1.
[2]
[PDF] Chapter 4: Maps between groups
The inner automorphism group. Definition. An inner automorphism of G is an automorphism ϕx ∈ Aut(G) defined by ... The outer automorphism group. Definition. An ...Missing: theory | Show results with:theory
[3]
[PDF] Homework Read sections 2.4, 2.5, 2.6, 2.7, 2.8 in Lauritzen's book ...
c) Show that the map g 7→ cg is a homomorphism from G to AutG, whose image is the inner automorphism group InnG and whose kernel is the center Z(G). Deduce that ...
[4]
Group theory basics - Harvard Mathematics Department
An automorphism of this form is called an ``inner automorphism''; all such automorphisms are ``trivial'' (the identity) if and only if G is a commutative group.
[5]
[PDF] Summary of Introductory Group Theory
Oct 28, 2024 · The inner automorphisms are good examples of nontrivial automorphisms of a group (at least, they are generally non-trivial for a non-Abelian ...
[6]
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 ...<|control11|><|separator|>
[7]
inner automorphism - PlanetMath
Mar 22, 2013 · An inner automorphism of a group G is an automorphism corresponding to conjugation by some element x in G.
[8]
4.3: Homomorphisms - Mathematics LibreTexts
Dec 11, 2024 · Hence $\sigma_a: G \mapsto G$ is an isomorphism. Definition: An inner automorphism of a group $G$ is defined as, \(h_a: G \to G, a \in G ...
[9]
Groups - Abstract Algebra and Discrete Mathematics
One can use x inverse to produce the inverse of the inner automorphism based on x. Thus the inner automorphisms form a subgroup of all the automorphisms of G.
[10]
[PDF] Math 594. Solutions to Exam 1 1. (20 pts) Let G be a group. We ...
The image of this map is denoted Inn(G) and its elements are called the inner automorphisms of G.
[11]
[PDF] GROUPS ACTING ON A SET 1. Left group actions Definition 1.1 ...
. Example 3.11. Consider G = S3 = {e,(12),(13),(23),(123),(132)} and let it act on itself by conjugation as above. It is easy to see that (13)(12)(13) = (23) ...
[12]
[PDF] CONJUGATION IN A GROUP 1. Introduction A reflection across one ...
If G is a group with trivial center, then the group Aut(G) also has trivial center. Proof. Let ϕ ∈ Aut(G) and assume ϕ commutes with all other automorphisms.
[13]
[PDF] Groups - OSU Math
Nov 1, 2023 · ) For example, the quaternion group Q8 = {1, i, j, k, −1, −i ... Klein's 4-group. (vi) Now let |G| = 6, G = {1, a, b, c, p, q}. At ...<|control11|><|separator|>
[14]
G/Z(G) is Isomorphic to Inn(G) - Mathonline
G/Z(G) is Isomorphic to Inn(G) ... Proposition 1: Let G be a group. Then $G/Z(G)$ is isomorphic to $\mathrm{Inn}(G)$. Recall that $Z(G) = C_G(G)$ is the center of ...Missing: theory | Show results with:theory
[15]
[PDF] 19. Automorphism group of Sn - UCSD Math
The inner automorphism group of G, denoted Inn(G), is the subgroup of Aut(G) given by inner automor- phisms.
[16]
Characteristic subgroups of Aut(G) - Math Stack Exchange
Jun 5, 2015 · For a group G, let Aut(G) denote the group of all the automorphisms of G. Consider the following subgroups of Aut(G): Inn(G)=the group of ...Inn characteristic in Aut - Math Stack ExchangeI have a question about group $G$ which satisfies Inn$(G) $ char ...More results from math.stackexchange.com
[17]
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.
[18]
[PDF] About Automorphisms of Some Finite Groups
This thesis introduces group theory, focusing on automorphism groups of finite groups, and defines automorphisms as the focus of the paper.
[19]
CLASSIFYING THE FINITE SIMPLE GROUPS CHAPTER I
THEOREM. Every simple group has a solvable outer automorphism group. 2. Multiply transitive permutation groups. Likewise one can easily determine which simple ...
[20]
[PDF] On Groups with a Class-Preserving Outer Automorphism
Then Aut(G) acts naturally on the set of conjugacy classes of G, and we denote the kernel of this action by Autc(G). We refer to the elements of Autc(G) as ...
[21]
[PDF] Algebra Final Exam Solutions 1. Automorphisms of groups. (a) Define
An inner automorphism of G is an automorphism of the form αh = g 7→ hgh−1 for some h ∈ G. (b) Prove or disprove that every automorphism of S3 is inner.Missing: Inn( | Show results with:Inn(
[22]
[PDF] The Theory of p-Groups
We define an extraspecial group to be a p-group for which G/Z(G) is elementary abelian, and Z(G) has order p. Extraspecial groups appear frequently in ...
[23]
THE AUTOMORPHISM GROUP OF AN EXTRASPECIAL p-GROUP
The finite p-group P is called special if either. (i) P is elementary abelian or (ii) the center, commutator subgroup and Frattini subgroup of P all coincide ...
[24]
[PDF] A Survey on Automorphism Groups of Finite p-Groups - arXiv
Oct 25, 2006 · Finally the inner automorphism group is, of course, isomorphic to U/Z(U) ... Isaacs, Extensions of group representations over nonalgebraically.
[25]
None
Below is a merged summary of the sections from J.S. Milne's *Group Theory* notes (v4.01), combining all the information from the provided segments into a single, comprehensive response. To maximize detail and clarity, I will use a table in CSV format to organize the key concepts, their definitions, relationships, and relevant page references across the summaries. Following the table, I will provide additional narrative details and URLs that are not easily tabularized.
[26]
[PDF] Burnside's Theorem - Mathematics
Apr 3, 2010 · Therefore, the number of irreducible representations of G is equal to the number of conjugacy classes of G. Proof. We already know that {χV1 ...
[27]
[PDF] 3 p-groups - Brandeis
One of the most important properties of p-groups is that they have nontrivial centers: Theorem 3.1. Every nontrivial p-group has a nontrivial center (P 6= 1 ⇒.
[28]
ATLAS: Alternating group A5, Linear group L2(5), Linear group L2(4)
### Summary of Automorphism Group, Inner/Outer Automorphisms of A5
[29]
ATLAS: Monster group M
### Summary: Automorphism Group of the Monster Group M
[30]
[1106.3760] Automorphism groups of sporadic groups - arXiv
Jun 19, 2011 · Among the simplest invariants of the sporadic finite simple groups are their outer automorphism groups. For 12 of the 26 possible isomorphism ...
[31]
[PDF] Lecture 2 — Some Sources of Lie Algebras
Sep 14, 2010 · (b) The inner derivations of a Lie algebra g form an ideal of Der(g). More precisely, [D,ad(a)] = ad(D(a)) for all D ∈ Der(g) and a ∈ g.
[32]
[PDF] Lie Algebras - UCSD Math
Jan 13, 2014 · We can define the quotient Lie algebra L/kerθ and it is isomorphic to imθ. Definition 2.15. (2.1) The centre Z(L) is a Lie algebra defined as { ...
[33]
Centre of Lie Algebra $sl_2(\mathbb{F}) - Math Stack Exchange
Feb 6, 2015 · The centre Z of L=sl2(F) is an abelian ideal in L, different from L itself. Hence dim(Z)≤2. Suppose that dim(Z)=1 or 2.Lie algebras sl(2,R) and (R3,∧) are not isomorphicShowing the Lie Algebras su(2) and sl(2,R) are not isomorphic.More results from math.stackexchange.com
[34]
[PDF] Quaternion algebras - John Voight
Mar 20, 2025 · Quaternion algebras sit prominently at the intersection of many mathematical subjects. They capture essential features of noncommutative ...<|control11|><|separator|>
[35]
[PDF] 1. Introduction Let F be an algebraically closed field of characteristic ...
Since any automorphism of Mn(F) is inner we may identify. AutF Mn(F) with. PGLn(F) = GLn(F)/F×. 1 simple means simple over an algebraic closure F of F. 2. Skew4 ...Missing: M_n( | Show results with:M_n(
[36]
[PDF] This is the final preprint version of a paper which ... - Berkeley Math
Inner automorphisms and inner endomorphisms of groups. Recall that an automorphism α of a group G is called inner if there exists an s ∈ G such that α is.
[37]
[PDF] inner automorphisms of groupoids - Macquarie University
Jul 24, 2019 · In this note we show that, in the context of groupoids, extended inner auto- morphisms once again capture a natural notion of conjugation: not ...
[38]
[PDF] Computation of outer automorphisms of central-simple algebras
Abstract. The paper presents algorithms for extending a field automorphism to a given finite-dimensional central-simple algebra over that field and, more.