All Questions
Tagged with finite-groups automorphism-groups
37 questions
2
votes
0
answers
121
views
A-conjugately dense subgroup
A subgroup $H$ of a group $G$ is called conjugately dense in $G$ if $H$ has nonempty intersection with each class of conjugate elements in $G$. We know that if $G$ is finite, then $H=G$. Now, my ...
- 173
4
votes
2
answers
313
views
Structure of Sylow $p$-subgroup of $G$ with given property
Let $G$ be a finite group and $A\leq \operatorname{Aut}(G)$. Assume that $P$ is a Sylow $p$-subgroup of $G$ and $z\in Z(P)$ of order $p$ such that each subgroup of order $p$ of $P$ is $A$-conjugate ...
- 173
4
votes
1
answer
318
views
Can $\text{Aut}(G)$ be extended to contain $G$?
Let $G$ be a group (finite, say) with center $Z$. The automorphism group $\text{Aut}(G)$ sits in a short exact sequence
$$ 1 \to G/Z \to \text{Aut}(G) \to \text{Out}(G) \to 1. $$
So when $Z\neq 1$, as ...
- 815
2
votes
1
answer
345
views
A natural automorphism of a finite group with two generators?
I am looking for a proof or a counterexample for the following:
Let $G$ be a finite group generated by $f$ and $g$. The map $f\mapsto f^{-1}, g\mapsto g^{-1}$ can be extended to an automorphism $\...
- 131
4
votes
2
answers
253
views
Order of abelian subgroup of the automorphism group of an abelian group
Suppose $A$ is a finite abelian group and $B\leq\operatorname{Aut}(A)$ is abelian. Is it possible for the order of $B$ to be strictly greater than the order of $A$? What if I additionally impose that ...
- 1,338
2
votes
0
answers
134
views
Automorphisms of (nilpotent) groups : torsion cokernel on the abelianisation implies torsion cokernel on the center?
$\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\End{End}\DeclareMathOperator\GL{GL}\DeclareMathOperator\Z{Z}\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\Span{Span}\DeclareMathOperator\ker{ker}\...
8
votes
1
answer
320
views
Sylow $p$ of $\mathrm{Aut}(G)$ with $G$ finite simple?
I met the following problem:
Let $G$ be a finite simple group (non-commutative, otherwise trivial). Let $p$ be a prime number not dividing $|G|$. Prove that any Sylow $p$ subgroup of $\mathrm{Aut}(G)$ ...
- 709
2
votes
0
answers
120
views
Status of the automorphism tower problem for finite groups
This is problem 11.123 in the Kourovka notebook:
For a given group $G$, define the following sequence
of groups: $A_1(G) = G$, $A_{i+1}(G) = \operatorname{Aut}(A_i(G))$. Does there exist a finite ...
- 1,433
15
votes
1
answer
751
views
Finite abelian groups with fewer automorphisms than a subgroup
It is not too hard to find examples of finite groups which have fewer automorphisms than one of their subgroups. For example $\mathcal D_4 \times \mathbb Z/2\mathbb Z$ (where $\mathcal D_4$ is the ...
- 1,322
1
vote
0
answers
125
views
Is the commutator of the holomorph of generalized quaternion group abelian?
Let $Q_{2^{n}} = \langle x, y \mathrel\vert x^{2^{n-1}}=y^4 = 1, x^{2^{n-2}}=y^2, y^{-1}xy = x^{-1} \rangle$ be the generalized quaternion group of order $2^{n}$.
Let $\operatorname{Hol}(Q_{2^{n+1}})$ ...
- 19
5
votes
2
answers
640
views
Automorphism groups of simple groups of Lie type
$\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\PGL{PGL}$In “Automorphisms of finite linear groups”, Steinberg proves that any automorphism of a simple group of Lie type (normal or twisted) is a ...
- 71
8
votes
2
answers
501
views
On $p$-groups with abelian automorphism group
Let $G$ be a $p$-group of order $p^{n}\geq p^{7}$ and its automorphism group is elementary abelian $p$-group. Then, it is clear that $G$ is nilpotent of class $2$. However, the converse is not true in ...
- 463
6
votes
1
answer
587
views
What are the "simplest" polytopes with an automorphism group of $\mathrm M_{11} \hspace {-1.25pt} $?
Do any polytopes have an automorphism group of the smallest of the sporadic groups, the Matthieu group $\mathrm M_{11} \hspace {-1pt} $? Indeed, they must exist. What are the simplest such polytopes ...
- 179
1
vote
1
answer
317
views
Why $G/F(G)$ is isomorphic to a subgroup of ${\rm Out}(F(G))$?
I know two facts and I’ve managed to figure out how to prove one, but the other one is still a little confusing.
Let $G$ be a finite solvable group and $F(G)$ is the Fitting subgroup of $G$.
(1) $G/Z(...
user121195
2
votes
0
answers
158
views
On automorphism group
Let $p$ be a prime number, and let $\mathbb{F}_{p}$ be a finite
field of order $p$. Let $U_{n}$ denote the unitriangular group of $n\times n$ upper
triangular matrices with ones on the diagonal, over $...
- 181
9
votes
0
answers
445
views
Which finite solvable groups have solvable automorphism groups?
Is it possible to give a reasonable description of those finite solvable groups $G$ such that $A = {\rm Aut}(G)$ is also solvable?
The central case to deal with is that in which $G$ is a $p$-group of ...
- 44.4k
14
votes
1
answer
955
views
On the iterated automorphism groups of the cyclic groups
Let $C_n$ be the cyclic group of order $n$. Its automorphism group $Aut(C_n)$ is a group of order $\varphi(n)$ isomorphic to $(\mathbb{Z}/n\mathbb{Z})^{\times}$ the multiplicative group of integer ...
10
votes
1
answer
415
views
Small automorphism groups of groups
I do not know much about group theory, so sorry in case this question is not for MO.
For a finite group $G$, denote by $f(G)$ the number of elements of the automorphism group of $G$.
Question: For ...
- 26.5k
12
votes
2
answers
406
views
Does asymmetric fraction of finite groups tend to $0$?
Let’s define asymmetric fraction of a finite group $G$ as the number $$\mathrm{af}(G) = \frac{|\{(g, a) \in G \times \mathrm{Aut}(G)\mid a(g) = g\}|}{|G|\cdot|\mathrm{Aut}(G)|}.$$ Equivalently it can ...
- 2,618
3
votes
1
answer
252
views
How large can a symmetric generating set of a finite group be?
Let $G$ be a finite group of order $n$ and let $\Delta$ be its generating set. I'll say that $\Delta$ generates $G$ symmetrically if for every permutation $\pi$ of $\Delta$ there exists $f:G\...
- 33
7
votes
2
answers
1k
views
What are the automorphism groups of direct products of dihedral group D4
What is the automorphism group of direct sum of dihedral group of order $8$, $D_4$?
For example, $\mathrm{Aut}(D_4)$ is isomorphic to $D_4$. How about $\mathrm{Aut}(D_4\times D_4)$, $\mathrm{Aut}(...
- 81
4
votes
1
answer
635
views
Schreier conjecture -- without a simple proof? and sporadic simple groups
The Schreier conjecture asserts that $\mathrm{Out}(G)$ is always a solvable group when $G$ is a finite simple group. This result is known to be true as a corollary of the classification of finite ...
- 10.5k
12
votes
2
answers
1k
views
Graph automorphism group
Let $A_w$ denote such set of positive integer $n$ that: for any two permutations $\pi_0,\pi_1\in S_n$, if $\pi_1$ is not a power of $\pi_0$, then there exists a (labeled non oriented) graph $G$ of ...
- 909
14
votes
4
answers
697
views
Non-split Aut(G) $\to$ Out(G)?
I'm looking for examples of outer automorphisms of a finite group $G$ which do not lift to automorphisms (i.e. non-split quotient map $\mathrm{Aut}(G)\to \mathrm{Out}(G)$, where $\mathrm{Out}(G) = \...
- 12.8k
1
vote
1
answer
286
views
Automorphism group of a graph
Suppose $\Gamma$ is a simple graph and $G=\mathrm{Aut}(\Gamma)$ is the automorphism group of $\Gamma$. If $G$ stabilizes a subgraph $\Gamma_1$,, and $G_0$ is the point-wise stabiliser of the set $V(\...
- 81
7
votes
1
answer
544
views
Automorphism group of $UT(n,p)$, the group of unitriangular matrices over the field $\mathbb{F}_p$
I am interested in understanding (at least roughly, if no such a description exists) the group of automorphisms for the group $UT(n,p)$, of unitriangular matrices over the field $\mathbb{F}_p$ on $p$-...
- 441
8
votes
1
answer
485
views
The automorphism of a group with a given fixed point set
It is a very known fact that for every group $G$ and an automorphism $\sigma$, the set of fixed points of $\sigma$ is a subgroup of $G$. My question is about the converse.
For a finite group $G$ and ...
- 81
3
votes
1
answer
836
views
Groups with abelian automorphism group
In a paper, the authors Jonah-Konvisser say
Until recently (~1975), there were no published examples of non-abelian groups with abelian automorphism groups. Heinken and Liebeck have methods for ...
- 261
3
votes
2
answers
337
views
Frobenius Groups of Automorphisms
Recently, I am looking different papers on the topic
$$\mbox{Frobenius groups of automorphisms of a group.}$$
But I am familiar with Frobenius groups only; not with their (faithful) actions on groups. ...
- 261
12
votes
1
answer
1k
views
Find finite groups $G\cong Aut(G)$
I become interested in this problem because $G\cong Aut(G)$ suggests a special symmetry in $G$ (This kind of group describes its own symmetry).
From $G/Z(G)\cong Inn(G)$ we know complete group is the ...
- 155
9
votes
1
answer
3k
views
Automorphism group of a finite group
I would like to ask if there exists an explicit description of $\mathrm{Aut}(G)$, the group of automorphisms of a finite group $G$, in particular, when $G$ is abelian. E.g., if $G = \mathbb{Z}/m\...
- 675
17
votes
0
answers
512
views
Maximum automorphism group for a 3-connected cubic graph
The following arose as a side issue in a project on graph reconstruction.
Problem: Let $a(n)$ be the greatest order of the automorphism group of a 3-connected cubic graph with $n$ vertices. Find a ...
- 37.7k
7
votes
2
answers
620
views
Automorphisms of subgroup of hamming cube under distance constraint
Let $S$ be a subset of $\{0,1\}^n$ such that any two elements of $S$ are at least (Hamming) distance 5 apart. I'm looking for an upper bound on the size of the automorphism group of $S$.
There's a ...
- 71
7
votes
1
answer
3k
views
Sylow subgroups invariant under an automorphism
Let $G$ be a finite group and $\sigma$ an automorphism of $G$. Suppose $p$ is a prime and $\sigma$ has prime order $q \neq p$. It's easy to see that $\sigma$ fixes a Sylow $p$-subgroup of $G$ if $\...
- 97
2
votes
1
answer
778
views
Automorphism of a wreath product
Let $S_k \wr S_n$ be the wreath product of two symmetric groups (so $S_n$ acts on $(S_k)^n = S_k \times ... \times S_k$ by permuting the factors; we then take the semi-direct product).
What is $Aut(...
- 1,180
10
votes
1
answer
2k
views
Non-trivial consequences of Baer's theorem and Lucchini's theorem in subnormality theory
There are a couple of beautiful results in finite group theory that look trivial, at least on a first glance, but require non-trivial facts to prove. I am basically interested in whether these results ...
- 1,993
73
votes
4
answers
4k
views
Is ${\rm S}_6$ the automorphism group of a group?
The automorphism group of the symmetric group $S_n$ is $S_n$ when $n$ is not $2$ or $6$, in which cases it is respectively $1$ and the semidirect product of $S_6$ with the (cyclic) group of order $2$. ...
- 1,069