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Questions tagged [automorphism-group]

For finding, constructing and proving results about automorphisms and applying them into different contexts. An automorphism is an isomorphism from an object to itself, and collectively they form a group under composition of mappings. Sometimes the only automorphism is the trivial one (the identity map), but often structures will have interesting/non-trivial automorphisms.

31
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Order of automorphism group of cyclic group

Let $G$ be a cyclic group of order $m$. What is the order of $\text{Aut}(G)$? I want to know the proof as well (elementary if possible). I would still accept the proof if one answers with $m = p$, a ...
8
votes
2answers
157 views

Is there a way to describe all finite groups $G$ such that $\operatorname{Aut}(G) \cong S_3$?

Is there a way to describe all finite groups $G$ such that $\operatorname{Aut}(G) \cong S_3$? Two groups that definitely satisfy that condition are $S_3$ itself (as it is a complete group) and $\...
3
votes
1answer
71 views

Show that if $p$ divides $\lvert\operatorname{Aut}(G)\rvert$, then $p=4k+3$

Give an example of some odd prime number $p$ and some group $G$ such that $|G|=p+1$ and $p$ divides $\lvert\operatorname{Aut}(G)\rvert$. Let $p$ is an odd prime number and $G$ is a group where $|G|=p+...
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1answer
185 views

Please help: How to show that Aut(G) is a subgroup of Sym(G) [closed]

I have to prove that if G is a group, Aut(G) is a subgroup of Sym(G), and later that the inner automorphisms group is a normal subgroup of Aut(G). I have time until tomorrow in the morning, but I have ...
5
votes
1answer
339 views

Group isomorphic to its automorphism group

Any complete group (that is, with trivial center and outer automorphism group) is isomorphic to its automorphism group. The inverse is not true, as the dihedral group of order 8 is isomorphic to its ...
5
votes
1answer
130 views

Is the statement that $ \operatorname{Aut}( \operatorname{Hol}(Z_n)) \cong \operatorname{Hol}(Z_n)$ true for every odd $n$?

Is the statement that $ \operatorname{Aut}(\operatorname{Hol}(Z_n)) \cong \operatorname{Hol}(Z_n)$ true for every odd $n$? $Hol$ stands here for group holomorph. This problem appeared, when I ...
0
votes
1answer
76 views

Galois group of a quintic with 3 real roots. How to conclude that there's one cycle of order 5?

I understand perfectly the argument making use of Cauchy's theorem, which I'll lay down for clarity's sake: take $p(x)$ of degree 5 irreducible over $\mathbb{Q}$. Let $K$ be the root field of $p(x)$ ...
7
votes
1answer
141 views

Groups which can not occur as automorphism group of a group

Consider the following natural question: Given a finite group $H$, does there exists a finite group $G$ such that $\mathrm{Aut}(G)\cong H$? In short, does any finite group occurs as the ...
6
votes
1answer
124 views

Are all power automorphisms of a finite abelian group of the form $f(x)=x^n$?

An example of a universal power automorphism, that is automorphisms of the form $f(x)=x^n$, is easy to find in an abelian group. Any power $n$ will work as long it is relatively prime to the orders of ...
2
votes
2answers
73 views

When is the automorphism group of a finite $p$-group nilpotent?

Suppose $G$ is a finite $p$-group with odd $p$. Is it true, that $Aut(G)$ is nilpotent iff $G$ is cyclic? When $G$ is cyclic, $Aut(G)$ is indeed abelian and thus nilpotent. However, I do not know ...
0
votes
2answers
76 views

Degree of extension of fixed field by infinite set of automorphisms.

If $G$ is a finite group of automorphism $E \rightarrow E$, then Dedekind-Artin theorem tells us that $[E:E^G]=\; \mid G \mid$ where $E^G$ is the subfield of $E$ fixed by the automorphisms of $G$. Is ...
7
votes
1answer
136 views

What is an example of a power automorphism of a group that isn't a universal power automorphism?

A power automorphism maps every subgroup of a group to within itself, with equality if the group is finite. More specifically, for a subgroup $H$ of $G$, a power automorphism $f$ has $f(H) \subseteq H$...
1
vote
1answer
122 views

Is it useful to know that automorphisms on $(\mathbb R^{\gt0},+)$ are always continuous?

I find it interesting that any automorphism of the semigroup $(\mathbb R^{\gt0},+)$ is continuous. This is also true if we assume the Axiom of Choice; c.f. Automorphisms on (R,+) and the Axiom of ...
6
votes
1answer
97 views

Does every finite non-trivial complete group have even order?

Does every finite non-trivial complete group have even order? I checked three well known classes of complete groups, and this statement is true for them all: 1) Symmetric groups: All symmetric ...
2
votes
0answers
55 views

Relating the automorphisms of $P$ and those of $P/Φ(P)$

Below is part of an exercise in Section 6.1 of Abstract Algebra: Dummit and Foote. 26. (e) Let $p$ be a prime. Let $P$ be a finite $p$-group. Let $\sigma$ be any automorphism of $P$ of prime order $...
2
votes
1answer
60 views

How to show that an automorphism is inner/outer

For fixed $C \in \mathrm{\mathbf{GL}}_n(\mathbb{R})$, we defined $\varphi, \psi: \mathrm{\mathbf{SL}}_n(\mathbb{R}) \rightarrow \mathrm{\mathbf{SL}}_n(\mathbb{R})$ as $$\varphi(A) = CAC^{-1}, \quad \...
0
votes
0answers
37 views

Number of elements in the cyclotomic $F$ class.

Let $F$ be a finite field with $p^n$ elements with prime $p > 5$ and $G = S_5$ the symmetric group. Let $l$ be the l.c.m. of the orders of the elements of G and $\theta$ be the primitive $l^{th}$ ...