Linked Questions

53 votes
13 answers
6k views

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 ...
24 votes
8 answers
2k 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 ...
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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
745 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
12 votes
1 answer
981 views

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 $G$ ...
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12 votes
1 answer
486 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 ...
7 votes
1 answer
362 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 ...
9 votes
0 answers
299 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 ...
11 votes
0 answers
154 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, ...