# Tagged Questions

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### quasiprimitive unsoluble groups

I'm looking for the reference about quasiprimitive unsoluble groups. Actually we can find a lot of useful things about quasiprimitive solvable groups in "Representations of solvable groups by Manz and ...
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### Simple groups with cyclic odd Sylow subgroups

I know that every odd Sylow subgroups of $PSL(2,p)$ is cyclic. Is there any other simple group with cyclic odd Sylow subgroups. Thank you in advance
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### Examples of Infinite Simple Groups

I would like a list of infinite simple groups. I am only aware of $A_\infty$. Any example is welcome, but I'm particularly interested in examples of infinite fields and values of $n$ such that ...
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### The center of Sylow $p$-subgroups of a finite simple group of Lie type

Would some one please to introduce me an easy reference!! which contains the size of $Z(P)$(center of $P$), where $P$ is a Sylow $p$-subgroup of a finite group of Lie type over a finite field of ...
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### Sylow $p$-subgroups of finite simple groups of Lie type

I need some information about the Sylow $p$-subgroups, and their normalizers, (specially their sizes), of a finite simple group of Lie type over a finite field (not necessarily algebraic closed) of ...
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### regular unipotent elements

Let $G$ be a finite simple group of Lie type over a finite field of characteristic $p\neq 2$ and $q$ elements. We know that if $p$ is not bad for $G$ then $G$ contains $q^l$ regular unipotent elements ...
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### Simple non-abelian groups

Let $G$ be a group and $H$ be a simple non-abelian subgroup of $G$ which is ascendant in $G$. Is it true that $H$ is also subnormal in $G$? Definition Let $G$ be a group and $H$ be a subgroup of $G$. ...
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### Let $G$ be a finite simple group. Suppose that $H$ is a subgroup of $G$ with index $n=|G:H|>1$. Show that $|H|$ divides $(n-1)!$

Let $G$ be a finite simple group. Suppose that $H$ is a subgroup of $G$ with index $n=|G:H|>1$. Show that $|H|$ divides $(n-1)!$ Hint: consider the action of $G$ on right cosets of $H$ in $G$. I'm ...
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### How can I prove that $A_5$ is perfect?

I'm trying to prove that $A_5$ is perfect. The only proof I found until now is: "It follows from the fact that it's simple and non-abelian". Simplicity is quite stronger, and since I only need ...
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### Check a proof for a theorem about the number of normal subgroups of $G \times H$ when $G$ and $H$ are simple

I'm going to provide a detailed proof for this theorem: Let $G$ and $H$ be non-trivial simple groups. If there is a prime number $p$ with $$|G|=|H|=p$$ then $G\times H$ has $p+3$ normal ...
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### Showing that a group is simple.

Given a non-commutative group of order $\mathrm{ord}(G)=343$. Prove that $G$ is a simple group. So I have to show that the only normal subgroups are the trivial group and the group itself. But I ...
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### Existance of regular semisimple elements in a torus

I had asked a question about regular semisimple elements in finite simple group of Lie type in this link Existance of semisimple elements in a torus. Actually I wanted to know that is it true that ...
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### Why projective linear spaces are simple ??? [closed]

In classical groups, people project all those linear groups and make it as simple. Why do they so, Why can't linear groups can be restricted to simple instead of projecting on the scalar space. What ...
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### Sort-of-simple non-Hopfian groups

A finite simple group is one which has no homomorphic images apart from itself and the trivial group. However, the simple-groups tag does not include the condition "finite". My question is the ...
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### Non simplicity of a group of order $p^{100}q$ given some conditions.

Let $G$ be a group $p$, $q$ primes $p\neq q$, $|G|=p^{100}q$. 1.- Assume that $G$ has two different $p$-Sylow subgroups and intersection for each pair of $p$-Sylow subgroups is trivial. Show that ...
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### Criterion for being a simple group

In this work it's written that A group $G$ is simple if and only if the diagonal subgroup of $G \times G$ is a maximal subgroup. How can I prove it?
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### Centralizer/Normalizer of abelian subgroup of a finite simple group

My concern is to look for a classification of : Centralizers/Normalizer of elementary abelian subgroups of a simple group. Centralizers/Normalizer of abelian subgroups(Not necessarily elementary) ...
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### An exercise about simple groups

Let $G$ be a finite group, $D=\{(g,g):g\in G\}$ is a subgroup of the direct product $G\times G$. Show that $G$ is simple if and only if $D$ is a maximal subgroup of $G\times G$. I tried to prove by ...
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### Prove that $G=S_{1}\times S_{2}\times\cdots\times S_{k}$

Let $G$ be a finite group and let $N_1,N_2,\ldots,N_k$ be normal subgroups. Suppose that $\bigcap_{i=1}^{k}N_{i}=\{1\}$ and that $G/N_i=S_i$ is simple. If all the groups $S_i$ are non-isomorphic ...
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### Conjugacy classes of two elements of order $p$ in a simple group

Let $G$ be a non abelian simple group and $x,~y$ be two elements of order $p$ of $G$, where $p$ is a prime. Suppose $|x^G|\neq |y^G|$. Is there any relation between $|x^G|$ and $|y^G|$? For example is ...
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### Prove that if a finite solvable group is simple, it is a cyclic group of prime order. [closed]

Prove that if a finite solvable group is simple, it is a cyclic group of prime order. Help me some hints.
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### Groups of order $56$

Let $G$ be a group of order $56$. (We do NOT assume the Sylow-$7$ subgroup to be normal.) Then either the Sylow-$2$ subgroup is normal or the Sylow-$7$ subgroup is normal. How to prove? My idea: ...
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### Homomorphism defined on a simple group

Can anybody please help me with this question??? Let $G$ and $G'$ be groups with $G$ simple. Prove that if $|G| > |G'|$, then the only homomorphism $\varphi : G → G'$ is the trivial one. ...
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### A connection between nonplanar complete graphs and the alternating group?

I went to an undergrad's senior honors thesis presentation a few days ago. She was discussing crossing numbers and mentioned that complete graphs $K_n$ are nonplanar iff $n \geq 5$. ?Coincidentally? ...
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### Permutation representation argument validity

I would like to check if the work I have done for this problem is valid and accurate. Any input would be appreciated. Thank you. Problem statement: Let $G$ be a group of order 150. Let $H$ be a ...
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### Simple group with order $\geq n!$ cannot have subgroup of index $n$.

My problem is as seen in the title: For positive integer $n>1$, prove that a simple group with order $\geq n!$ cannot have subgroup of index $n$. Could anyone give me some hints on how to ...
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### Missing $\{2,p\}$-Hall subgroups in finite non-abelian simple groups

Can anybody tell me how to prove this theorem? Theorem: Suppose that $G$ is a finite non-abelian simple group. Then there exists an odd prime $p\in\pi(G)$ such that $G$ has no $\{2,p\}$-Hall ...
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### If $G_1 \leq G_2 \leq \cdots \leq G$ such that $G = \bigcup_{i=1}^\infty G_i$ and each $G_i$ is simple then $G$ is simple.

Prove that if there exists a chain of subgroups $G_1 \leq G_2 \leq \cdots \leq G$ such that $G = \bigcup_{i=1}^\infty G_i$ and each $G_i$ is simple then $G$ is simple. I have proved the following ...
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### Verifying finite simple groups

The classification of finite simple groups required thousands of pages in journals. The end result is that a finite group is simple if and only if it is on a list of 26 sporadic groups and several ...
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### Who did first use the term “simple” in group sense and why?

Who did first use the term "simple" in group sense? More appreciated if I learn the original word(might be in French, Galois???)... and also... Why do you think that this term is chosen ...
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### Are quasinilpotent groups a Fitting class?

A finite group is called quasinilpotent if it induces inner automorphisms on all of its chief factors. A solvable group is quasinilpotent iff it centralizes all of its (necessarily abelian) chief ...
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### Are there any distinct finite simple groups with the same order?

In finite group theory, it's often quite easy to show that there are no simple groups of a given order $n$. My question is different: is there some natural number $n$ such that there are two ...
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### Is there a simple group of any (infinite) size?

I'm trying to show that for any infinite cardinal $\kappa$ there is a simple group $G$ of size $\kappa$, I tried to use the compactness theorem and then ascending Löwenheim-Skolem, but this is ...
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### Are the quotient groups in a composition sequence necessarily subgroups?

Does there exist a finite group G and a normal subgroup N of G so that G/N is a simple group and G/N is not isomorphic to any subgroup of G ?
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### Simple groups some interesting properties

I have found some interesting results as follows: If $o(G)\not|\ i(H)!$ then $H$ contains a non-trivial normal subgroup of $G$, where $i(H)=[G:H]$. 2.If $o(G)=2m$, where m is an odd prime number ...
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### The Monster group

I do know that there is a "long proof" (I have read that it is +1000pages) that the Monster group is the largest sporadic simple group. My question is: Why can we be sure that there is no other ...
Let $q\geq 5$ and let PGL(2,q) be the projective general linear group. Question Do there exists a $q$ such that PGL(2,q) is solvable?
### Subgroups of $A_5$ have order at most $12$?
How does one prove that any proper subgroup of $A_5$ has order at most $12$? I have seen that there are $24$ $5$-cycles and $20$ $3$-cycles. What do the other members of $A_5$ look like?