Linked Questions
12 questions linked to/from Does $\DeclareMathOperator\Aut{Aut}\Aut(\Aut(\dots\Aut(G)\dots))$ stabilize?
59
votes
13
answers
8k
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Cardinalities larger than the continuum in areas besides set theory
It seems that in most theorems outside of set theory where the size of some set is used in the proof, there are three possibilities: either the set is finite, countably infinite, or uncountably ...
- 5,739
24
votes
8
answers
3k
views
Applications of logic to group theory?
There seems to be an ever-growing literature on the first-order theory of groups. While I find this interaction between group theory and logic quite appealing, I was wondering the following:
Are ...
- 359
11
votes
4
answers
3k
views
Is there a non-trivial group G isomorphic to Aut(G)?
The title basically says it all.
Is there a group with more than one element that is isomorphic to the group of automorphisms of itself?
I'm mainly interested in the case for finite groups,
although ...
user5810
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
33
votes
1
answer
823
views
Two groups that are the automorphism groups of each other
Let $H,K$ be two non-isomorphic groups such that $H\cong Aut(K)$ and $K\cong Aut(H)$.
Is there any example of such groups ?
Note: I had asked the question there.
- 1,169
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 ...
13
votes
1
answer
1k
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Iterated Automorphism Groups
Notation: For each group $G$ define:
$Aut^{(0)}(G):=G$
$Aut^{(1)}(G):=Aut(G)$
$\forall n\geq 1~~~Aut^{(n+1)}(G):=Aut(Aut^{(n)}(G))$
Question: Consider $I\subseteq \omega$. Is there a group $G$ ...
user43940
8
votes
1
answer
505
views
Conditions for a finite group to be isomorphic to its automorphism group
So in the interest of gaining a better understanding of a conjecture (due to Scott, 1960) on the automorphism series (first part of the automorphism tower, no direct limits) of a finite group that ...
- 461
6
votes
1
answer
242
views
Results with a flavor “every automorphism of automorphisms is inner”
It seems that there are a number of results which take more or less the following form: let $X$ be some (specific) kind of structure, let $Y$ be the group of automorphisms of $X$ or perhaps ring of ...
9
votes
0
answers
372
views
Groups with trivial outer automorphism group and prescribed center?
Given an arbitrary abelian group $A$, can we find a group $G$ such that
$\mathrm{Out}(G)=\mathrm{Aut}(G)/\mathrm{Inn}(G)=1$, and
$Z(G)\simeq A$?
Why is this interesting? Given a group $G$, we have ...
- 27.2k
11
votes
0
answers
186
views
Iterated automorphism groups of finite groups
Let $\mathcal{G}$ be the set of isomorphism classes of finite groups.
There is an operation $\mathrm{Aut} : \mathcal{G} \rightarrow \mathcal{G}$ which gives the automorphism group of a given group, ...
- 12.4k
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