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Questions tagged [frobenius-groups]

Use this tag for questions about Frobenius groups, kernels and complements.

A Frobenius group is a transitive permutation group on a finite set, such that no non-trivial element fixes more than one point and some non-trivial element fixes a point. Alternatively, G is a Frobenius group if and only if G has a proper, nonidentity subgroup H such that H ∩ Hg is the identity subgroup for every g ∈ G − H, i.e. H is a malnormal subgroup of G.

51 questions
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Galois group of the splitting field of the polynomial $x^5 - 2$ over $\mathbb Q$

We know that the splitting field of $x^5 - 2 $ over $\mathbb Q$ is $\mathbb Q(2^{1/5}, \rho)$, where $\rho$ is a fifth root of unity. Therefore, $\left[\mathbb Q(2^{1/5} , \rho) : \mathbb Q \right] = 20 $. Let $G$ be the Galois group of $\mathbb…
Struggler
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Frobenius kernel is regular normal elementary abelian p-subgroup?

I'm attempting Exercise 3.4.6 in Dixon & Mortimer's book on Permutation Groups: Let $G$ be a finite primitive permutation group with abelian point stabilisers. Show that $G$ has a regular normal elementary abelian $p$-subgroup for some prime…
M.Hope
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Does every $p$-group of odd order admit fixed point free automorphisms?

Does every $p$-group of odd order admit fixed point free automorphisms? equivalently, Given an odd order $p$-group $P$, is there a group $C$ such that we can form a Frobenius group $P\rtimes C$? Note that this is not true for $p$-groups of even…
Samuel Handwich
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What are the Frobenius groups of order $100$?

Question. Which groups of order $100$ are Frobenius groups? The OEIS says that there are two Frobenius groups of order $100$, but I am finding three of them and I'd be grateful if someone can point out where I am going wrong. (I was not able to…
James
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On groups of order $p^aq^br^c$ containing elements of order $pq$, $qr$, and $pr$, but not $pqr$

Let $G$ be a solvable group of order $p^aq^br^c$ (for distinct prime $p,q,r$) containing elements of orders $pq$, $qr$, and $pr$, but no element of order $pqr$. Furthermore, assume that $G$ is minimal with respect to this property, i.e. any proper…
Samuel Handwich
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Subgroup containing normalizers of its $p$-subgroups

This is exercise $1.D.5$ of Isaacs' "Finite Group Theory". It goes: Let $G$ be a finite group and let $H$ be a subgroup of $G$. Suppose $N_G(P) \subset H$ for all $p$-subgroups $1 \neq P$ of $H$. Then, $H$ is a Frobenius complement in $G$. For the…
Gauss
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If $G$ is nonabelian & solvable s.t. the centralizer of each nontrivial element is abelian, then $G$ is Frobenius with kernel its Fitting subgroup

I'm dealing with the following problem in Isaacs Finite Group Theory [6A.5], I would appreciate if you could help: Let $G$ be a nonabelian solvable group in which the centralizer of every nonidentity element is abelian. Show that $G$ is a Frobenius…
Yılmaz
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Gorenstein's proof of the classification of solvable CN-groups

I am reading Gorenstein's Finite Groups. Chapter 14 is about CN-groups, a (finite) group where the centralizer of every non-identity element is nilpotent. Theorem 14.1.5 gives the classification of solvable CN-groups, which states that any such…
Ted
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on primitive group actions with abelian stabilizers

I am trying to solve the following exercise from Dixon and Mortimer: Let $G$ be a finite primitive permutation group with abelian point stabilizers. Show that $G$ has a regular normal elementary abelian subgroup. As hints the authors suggest to…
Dune
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Order of a product of two elements in a Frobenius Group

Let $G = K \rtimes H$ be a Frobenius group with Kernel $K$ and complement $H$. I would like to show that $$ o(xy) = o(y), \quad \mbox{for every} \ x \in K \ \mbox{and for every} \ y \in H \setminus \{1\} $$ Since $K \trianglelefteq G$ and $K \cap H…
Rodolfo Ferreira de Oliveira
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About Frobenius Groups of order 1029

In the list of groups in GAP of order $1029=7^3\cdot 3$, there are two, with structure description $U_3(\mathbb{F}_7)\rtimes C_3$. (Among $19$ groups $G[1], G[2], \ldots, G[19]$ of order $1029$, the groups $G[11]$ and $G[13]$ have structure…
Maths Rahul
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$G$ a group with center $\{e\}$ and $A$ a maximal subgroup of $G$ that is also abelian and not normal. How to show that $A$ is a Frobenius complement?

I have been sitting on this homework problem for days now: As the title says, I have a group (which doesn't have to be finite. Even Frobenius groups aren't defined as finite in our course) which only has the neutral element $e$ as center. $A$ is a…
joern
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A question about Frobenius action

On page 177 of my textbook Finite Group Theory by I. Martin Isaacs, it says: Let $A$ and $N$ be finite groups, and suppose that $A$ acts on $N$ via automorphisms. The action of $A$ on $N$ is said to be Frobenius if $n^a\not=n$ whenever $n\in N$ and…
user517681
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Frobenius Mapping as a Ring Homomorphism

I'm studying Lang's Algebra, where it is observed that, for the finite field $\mathbf{F}_q$ with $q=p^n$ elements ($p$ prime), the Frobenius mapping $$ z\mapsto z^p$$ is a ring homomorphism with kernel 0. This doesn't seem obvious to me. Why is the…
0-seigfried
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Why the normalizer of the Sylow $p$-subgroups of the symmetric group of degree $p$ has order $p(p-1)$ and is known as Frobenius group $F_{p(p-1)}$?

Why the normalizer of the Sylow $p$-subgroups of the symmetric group of degree $p$ has order $p(p-1)$ and is known as Frobenius group $F_{p(p-1)}$? I am trying to understand the statements on Wikipedia about Sylow subgroups of the symmetric…
Bach
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