A subgroup of a group is characteristic in it if every automorphism of the whole group takes the subgroup within itself. To be used with the tag [group-theory].
Questions tagged [characteristic-subgroups]
66 questions
12
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Reference request for characteristic subgroups of free abelian groups
The characteristic subgroups of a free abelian group $F$ are the $k F$ with $k \in \mathbb{N}$ . (In particular, they are verbal and hence fully invariant.) I know a proof when $F$ has finite rank, but I would like to know a reference of this fact…
Martin Brandenburg
- 183,887
10
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1 answer
Commutator subgroup $G'$ is a characteristic subgroup of $G$
For any group $G$, prove that the commutator subgroup $G'$ is a characteristic subgroup of $G$. Let $U=\{xyx^{-1}y^{-1}|x, y \in G\}$. Now $G'$ is the smallest subgroup of $G$ which contains $U$. We need to show that $T(G') \subset G'$ for all…
tattwamasi amrutam
- 13,130
- 5
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- 79
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votes
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$G$ is characteristically simple $\iff$ there is simple $T$ such that $G \cong T\times T \times \cdots \times T$
Let $G$ be a characteristically simple finite group, i.e. it has no nontrivial characteristic subgroups. Prove there is some simple group $T$ such that $G \cong T \times T \times \cdots \times T$. No idea how to start this one. I have tried to…
Spencer Hyman
- 374
7
votes
1 answer
Automorphism on subgroup lift to the whole group
By definition, if $H$ is characteristic subgroup of $G$ , then every automorphism of $G$ restricted on $H$ is an automorphism of $H$ . From these two questions (1) (2) we know it's generally false that if $H$ is characteristic subgroup of $G$ ,…
Andrews
- 4,403
6
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1 answer
If we have a Free Group with a chain of proper characteristic subgroups, then their intersection is trivial
We have $F$ a free group. If we have a chain $F = C_0 \gt C_1 \gt \ldots$ such that each $C_{i+1}$ is a proper characteristic subgroup of $C_i$ . We want to prove $\cap_i C_i = 1$ . Now, I have a feeling that we must use the fact that a free group…
Cactus
- 115
6
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3 answers
Autocommutator subgroup is a characteristic subgroup
Autocommutator subgroup $K(G)$ of a group $G$ is defined as $$K(G):=\langle [g,\alpha]:g\in G,\; \alpha \in Aut(G)\rangle .,$$ where $[g,\alpha ]:=g^{-1}\alpha (g)$ . Is $K(G)$ a characteristic subgroup in $G$ ? Recall that a subgroup $H$ of…
Mahtab
- 747
6
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2 answers
On group quotient by a characteristic subgroup
My question is: Let $H,K\leq G$ be two characteristic subgroups and assume $H\leq K$ . Do we have $K/H$ is characteristic in $G/H$ ? We know that any characteristic subgroup of $G/H$ must be of the form $K/H$ for some characteristic subgroup $K$ of…
abvdd
- 678
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3 answers
A simple question on characteristic subgroups
Suppose $P$ is a finite abelian $2$-group and $U$,$V$ are characteristic subgroups of $P$ such that $|V:U|=2$. Does it follow that $P$ has a characteristic subgroup of order $2$?
the_fox
- 5,978
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Examples of "unexpected" characteristic subgroups
If $A$ is an abelian group, there are a number of characteristic subgroups of it that immediately come to mind: Any of the groups $nA$ , $A[n]$ (the $n$ -torsion), the full torsion subgroup $T(A)$ , the maximal divisible subgroup $D(A)$ etc. Any of…
Tim Seifert
- 2,411
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Does there exist an Artinian verbally simple group, which is not characteristically simple?
Does there exist an Artinian verbally simple group, which is not characteristically simple? A characteristically simple group is a group without non-trivial proper characteristic subgroups, a verbally simple group is a group without non-trivial…
Chain Markov
- 16,122
5
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1 answer
Does there exist some sort of classification of finite verbally simple groups?
Let’s call a group verbally simple if it does not have any non-trivial verbal subgroup. Does there exist some sort of classification of finite verbally simple groups? $G^n$ , with $G$ being a finite simple group, is always verbally simple as it has…
Chain Markov
- 16,122
5
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1 answer
An example to show if $K$ is normal in $H$, $H$ is characteristic in $G$ then $K$ is not normal in $G$
I am looking for an example of statement " $K$ is normal in $H$, $H$ is characteristic in $G$ then $K$ is not normal in $G$ ". As for statement "$K$ is normal in $H$, $H$ is normal in $G$, $K$ not necessarily normal in $G$" I found out an example of…
4
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2 answers
Normal subgroup of a characteristic subgroup
I came across the following question while reviewing for my qualifying exams: Prove or provide a counterexample: If $M$ is a normal subgroup of $N$ , and $N$ is a characteristic subgroup of $G$ , then $M$ is a normal subgroup of $G$ . Looking at our…
slowspider
- 1,135
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Is my proof correct?: Let $K$ be a normal subgroup of a finite group $G$ and assume $|K|$ and $|G/K|$ are coprime. Prove $K$ is characteristic in $G$.
Let $f \in \mathrm{Aut}(G)$ . Let $f(K)=H$ . Consider $\pi: G \to G/K$ . Since $\pi(H)=HK/K \le G/K$ , then $|\pi(H)| \mid |G/K|$ . Also, since $HK/K \cong H/(H\cap K)$ , then $|\pi(H)|=|H/(H\cap K)|$ . So, $|\pi(H)||H\cap K|=|H|=|f(K)|=|K|$ . So,…
user5826
- 12,634
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3 answers
Normal subgroups vs characteristic subgroups
It is well known that characteristic subgroups of a group $G$ are normal. Is the converse true?
Louis La Brocante
- 2,147