2
votes
1answer
39 views

Equivalence classes of (2,3)-pairs in PSL(2,q)

Let $G = PSL(2,q)$. I am interested in the different ways of writing $G$ as an image of the modular group. If $A, B \in G$ are elements of order $2, 3$ respectively, then $\langle A,B \rangle$ is an ...
1
vote
1answer
64 views

The Diagonal Subgroup of $A \times A$ is Maximal iff $A$ is Simple

Let $A$ be a group and $G = A \times A$. Define $D= \{(a,a,)\mid a \in A\}$ (the diagonal subgroup of $G$). Prove that $D$ is a maximal subgroup of $G$ if and only if $A$ is simple, i.e. it has no ...
3
votes
1answer
46 views

There are no simple groups of order $27p$

Statement Show that there are no simple groups of order $27p$ for any $p$ prime number. I got stuck with this problem, I'll write what I've done so far: Suppose $G$ is a group with $|G|=3^3p$, $p$ ...
1
vote
1answer
75 views

How do we know the classification of finite simple groups is finished?

I don't have much experience in group theory beyond knowing what a finite simple group is, as well as the other basic parts of the topic, so far as I can tell, so if the actual answer is too complex ...
2
votes
1answer
64 views

Elements of $\operatorname{Aut}(\operatorname{Aut}(G))$ acting as an identity on $\operatorname{Inn}(G)$, for $G$ a nonabelian, simple group

Let $f$ be an element of $\operatorname{Aut}(\operatorname{Aut} G)$ s.t $f(r)$ is equal to $r$ for all $r \in \operatorname{Inn}(G)$. Prove that $f(j)$ is equal to $j$, $\forall ...
0
votes
0answers
23 views

finite simple group of Lie type

let $G$ be a finite simple group of Lie type. would someone please help me! I want to know, is the following expression true or not? " There exists a simple linear algebraic $\mathcal{G}$ over an ...
0
votes
1answer
88 views

How to prove that every nonabelian group of odd order is not simple?

How to prove or disprove that every nonabelian group of odd order is not simple? I have no ideas concerning that.
2
votes
1answer
91 views

Projective special linear groups are co-hopfian?

Is it known if $PSL(2,\, F)$, with $F$ a field of prime $p$ characteristic (maybe with all proper subfields of finite order), is co-hopfian? I've searched everywhere but found nothing. Definition A ...
0
votes
0answers
25 views

quasiprimitive non-solvable 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 ...
3
votes
1answer
72 views

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
3
votes
4answers
131 views

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 ...
0
votes
1answer
41 views

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 ...
2
votes
2answers
95 views

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 ...
1
vote
1answer
35 views

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 ...
1
vote
1answer
52 views

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 $G$. ...
2
votes
1answer
48 views

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 ...
3
votes
1answer
49 views

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 ...
1
vote
1answer
63 views

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 ...
0
votes
1answer
67 views

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 ...
0
votes
0answers
41 views

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 ...
-1
votes
1answer
44 views

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 ...
5
votes
1answer
88 views

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 ...
3
votes
3answers
113 views

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 ...
5
votes
1answer
75 views

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?
1
vote
1answer
55 views

Number of Sylow $5$-subgroups

Suppose $G=SL(2,4)$. Could you please suggest, is there is any simple argument to show that $G$ cannot have the unique Sylow $5$-subgroup, i.e. $|Syl_5(G)|>1$ without quoting the isomorphism $A_5 ...
1
vote
1answer
34 views

Simple subgroup of transitive group

Suppose $Q \in Syl_n(G)$ and $Q$ is not normal in $G$, $Q$ is generated by an element of order $n$, where $G \leqslant S_n$ for prime $n$ and $G$ acts transitively on $\{1,...,n\}$. Define H to be a ...
2
votes
1answer
66 views

How to show that group of order 760 is not simple? [closed]

How to show that group of order 760 is not simple? By Sylow's theorem $n_{19}=20$, and $o(N(P)) = 38$, but how to continue after this? Thanks for any help
2
votes
1answer
50 views

Is there a simple group and a proper subgroup with a unique and equidistant intermediate?

Let $G$ be a simple group and $H$ a proper subgroup such that there is a unique intermediate subgroup $K$ (i.e. $H < S < G$ implies $S=K$). Question: Is it possible that $[G:K]=[K:H]$ ? ...
5
votes
2answers
80 views

If $G$ is finite simple and $H \le G$ has index $2m$, then an involution of $G$ is conjugate to one in $H$

I have been trying to prove the following for a while now, with plenty of attempts and ideas, but no success. I would appreciate a gentle nudge in the right direction. Suppose that $G$ is a finite ...
0
votes
1answer
86 views

Normalizer of Sylow p subgroup

I am trying to prove a simple group of order 168 has no subgroup of order 14, the hint ask me show how many Sylow 7 subgroup it has and what is the order of the normalizer of such Sylow 7 subgroup. I ...
0
votes
0answers
33 views

Brauer characters of finite simpl groups of type E8

I would like to know the Brauer characters of finite simple groups $E_8(2)$, $E_8(3)$ or $E_8(5)$. Is there any refrence for this topic? Thanks
3
votes
1answer
97 views

Centralizer of unipotent element of simple groups

Let $G$ be a simple group of lie type over a field of odd characteristic $p$, and $u$ is a regular unipotent element of $G$, that is a $p$-element with centralizer of minimum rank. We know that ...
3
votes
1answer
83 views

Is $\mathrm{PSL} ( 2, \mathbb{Q} )$ a simple group?

I am a new poster but I don't think this question has been asked before. Pardon me if it is.
3
votes
2answers
193 views

Centralizer of involutions in simple groups.

I need some information about centralizer of involutions in finite simple groups of lie type. Actually I want to know if $G$ is a simple group of lie type over a finite field, How many conjugacy ...
0
votes
1answer
46 views

A question about nonabelian finite simple groups

Let $G$ be a nonabelian finite simple group of lie type on finite field $F$ and $s\in G$ be a semisimpl element of $G$, $i.e.$ an element with order coprime to $Char(F)$. Also suppose $T$ is a maximal ...
2
votes
2answers
48 views

Are there any nontrivial ways to factor n-cycles into a product of cycles?

I was reading a proof here about the simplicity of $A_n (n \ge 5)$. It states (and proves) a lemma about 3-cycles: A 3-cycle $(a, b, c)$ may be written as $(a, b, c) = (1, 2, a)^{-1}(1, 2, c)(1, 2, ...
3
votes
2answers
173 views

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) ...
4
votes
2answers
166 views

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 ...
4
votes
1answer
126 views

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 ...
3
votes
1answer
79 views

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 ...
-1
votes
2answers
135 views

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.
2
votes
2answers
98 views

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: ...
1
vote
1answer
70 views

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. ...
7
votes
0answers
97 views

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? ...
2
votes
0answers
42 views

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 ...
0
votes
1answer
85 views

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 ...
6
votes
0answers
58 views

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 ...
2
votes
2answers
69 views

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 ...
8
votes
2answers
181 views

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 ...
3
votes
2answers
113 views

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 ...